light mediator. For a tree-level interaction between DM and the SM via some mediator with mass $M_{\text{med}}$, the EFT approximation corresponds to expanding the propagator for the mediator in powers of $Q_{\text{tr}}^2/M_{\text{med}}^2$, truncating at lowest order, and combining the remaining parameters into a single parameter $M_*$ (connected to the scale of the interaction $\Lambda$ in the literature). For an example scenario with a Z'-type mediator (leading to some combination of operators D5 to D8 in the notation of [23] for the EFT limit), this corresponds to setting

$$ \frac{g_{\chi} g_{\text{q}}}{Q_{\text{tr}}^2 - M_{\text{med}}^2} = - \frac{g_{\chi} g_{\text{q}}}{M_{\text{med}}^2} \left( 1 + \frac{Q_{\text{tr}}^2}{M_{\text{med}}^2} + \sigma \left( \frac{Q_{\text{tr}}^4}{M_{\text{med}}^4} \right) \right) \approx - \frac{1}{M_*^2}, \quad (45) $$

where $Q_{\text{tr}}$ is the momentum carried by the mediator, and $g_{\chi}$, $g_{\text{q}}$ are the DM-mediator and quark-mediator couplings respectively.<fnref n="16" /> A minimal condition that must be satisfied for this approximation to be valid is that $Q_{\text{tr}}^2 < M_{\text{med}}^2 = g_{\chi}g_{\text{q}}M_*^2$. This requirement avoids the regions: $Q_{\text{tr}}^2 \sim M_{\text{med}}^2$, in which case the EFT misses a resonant enhancement, and it is conservative to ignore this enhancement; and $Q_{\text{tr}}^2 \gg M_{\text{med}}^2$, in which case the signal cross section should fall according to a power of $Q_{\text{tr}}^{-1}$ instead of $M_{\text{med}}^{-1}$. The latter is the problematic kinematic region.

The condition $Q_{\text{tr}}^2 < M_{\text{med}}^2 = g_{\chi}g_{\text{q}}M_*^2$ was applied to restrict the kinematics of the signal and remove events for which the high-mediator-mass approximation made in the EFT would not be reliable. This leads to a smaller effective cross-section, after imposing the event selection of the analysis. This truncated signal was then used to derive a new, more conservative limit on $M_*$ as a function of $(m_{\chi}, g_{\chi}g_{\text{q}})$.

For the example D5-like operator, where the cross section $\sigma$ scales as $M_*^{-4}$, there is a simple rule for converting a rescaled cross section into a rescaled constraint on $M_*$, if the original limit is based on a simple cut-and-count procedure. Defining $\sigma_{\text{EFT}}^{\text{cut}}$ as the cross section truncated such that all events pass the condition $\sqrt{g_{\chi}g_{\text{q}}}M_*^{\text{rescaled}} > Q_{\text{tr}}$, we have

$$ M_*^{\text{rescaled}} = \left( \frac{\sigma_{\text{EFT}}}{\sigma_{\text{EFT}}^{\text{cut}}(M_*^{\text{rescaled}})} \right)^{1/4} M_*^{\text{original}}, \quad (46) $$

which can be solved for $M_*^{\text{rescaled}}$ via either iteration or a scan. Similar relations exist for a given UV completion of each operator.

This procedure has been proposed in Ref. [145] and its application to ATLAS results can be found in Ref. [7] for a range of operators. We reiterate: knowledge of the UV completion for a given EFT operator was necessary for this procedure; this introduces a model-dependence that was not present in the non-truncated EFT results.

Currently, simplified models (including the full effect of the mediator propagator) are available for comparison with the data, and since knowledge of the simplified models is needed for the truncation procedure, there is no reason to apply this prescription. Instead, the simplified model limit for large $M_*$ can be presented for interpretation in terms of EFT operators.

### 5.1.2. EFT truncation using the center of mass energy

The procedure presented in the previous section was predicated on some knowledge of the simplified model. This led to the identification of the mass of the DM pair as the relevant kinematic quantity to use in a truncation procedure. In general, if no assumption is made about the underlying dynamics, it is more conservative to place a limit on the total center of mass energy $E_{\text{cm}}$ of the DM production process. Furthermore, the direct

connection between the mass scale of the EFT validity, $M_{\text{cut}}$, and the mass scale that normalizes the EFT operator, $M_*$, is unknown. For such cases, Refs. [97,146] proposed a procedure to extract model independent and consistent bounds within the EFT that can be applied to any effective Lagrangian describing the interactions between the DM and the SM. This procedure provides conservative limits that can be directly reinterpreted in any completion of the EFT. The condition ensuring that the EFT approximation is appropriate is:

$$ E_{\text{cm}} < M_{\text{cut}}. \quad (47) $$

The relationship between $M_{\text{cut}}$ and $M_*$ can be parameterized by an effective coupling strength $g_*$, such that $M_{\text{cut}} = g_* M_*$. A scan over values of $g_*$ provides an indication of the sensitivity of the prediction to the truncation procedure. In the Z'-type model considered above, $g_*$ is equal to $\sqrt{g_{\chi}g_{\text{q}}}$. The resulting plots are shown in [146] for a particular effective operator.

The advantage of this procedure is that the obtained bounds can be directly and easily recast in any completion of the EFT, by computing the parameters $M_*$, $M_{\text{cut}}$ in the full model as functions of the parameters of the complete theory. On the other hand, the resulting limits will be weaker than those obtained using $Q_{\text{tr}}$ and a specific UV completion.

### 5.1.3. Truncation at the generator level

The conditions on the momentum transfer can also be applied directly at the generator level, by discarding events that are invalid and calculating the limits from this truncated shape. This provides the necessary rescaling of the cross section while keeping the information on the change in the kinematic distributions due to the removal of the invalid events. This procedure is more general with respect to rescaling the limit in the two sections above, and it should be followed if a search is not simply a counting experiment and exploits the shapes of kinematic distributions.

### 5.1.4. Sample results of EFT truncation procedures

An example of the application of the two procedures to the limit on $M_*$ from Ref. [14] as a function of the product of the couplings is shown in Fig. 69. Only the region between the dashed and the solid line is excluded. It can be seen that the procedure from [146] outlined in Section 5.1.2, shown in blue, is more conservative than the procedure from Refs. [7,145], described in Section 5.1.1.

### 5.1.5. Comments on unitarity considerations

A further consideration applicable to EFT operators at hadron colliders is the potential violation of unitarity. An analysis of the operator $\frac{\bar{q}\gamma^\mu q\tilde{\chi}\gamma^\mu\chi}{M_*^2}$ provides the limit:

$$ M_* > \beta(s)\sqrt{s}\sqrt{\frac{\sqrt{3}}{4\pi}}, \quad (48) $$

where $\sqrt{s}$ is (maximally) the collider energy and $\beta(s)$ is the DM velocity [24]. Constraints for other operators have also been derived [147]. This constraint on $M_*$ still is open to interpretation, since the relation to $M_{\text{cut}}$ is not resolved, except for a specific simplified model. Derived limits on $M_*$ should be compared to this unitarity bound to check for consistency.

## 5.2. Recommendation for presentation of EFT results

In this report, we make two recommendations for the presentation of collider results in terms of Effective Field Theories for the upcoming Run-2 searches. A full discussion of the presentation of

<footnote n="16">¹⁶ Here, we ignore potential complications from the mediator width when the couplings are large.</footnote>
<figure><img src="image_93.png" /><figcaption>**Fig. 69.** 95% CL lower limits on the scale of the interaction of the D5 operator at 14 TeV, after the two truncation procedures. The procedure from [146] outlined in Section 5.1.2 is shown in blue, while the procedure from Refs. [7,145], described in Section 5.1.1 is shown in red. Only the region between the dashed and the solid lines is excluded. Even though the intersection between the two lines is not shown in this plot, it should be noted that no limit can be set anymore for sufficiently low couplings, whatever truncation method is used.</figcaption></figure>

collider results in relation to other experiments is left to work beyond this Forum, where ATLAS, CMS, the theory community and the Direct and Indirect Detection communities are to be involved.

We divide the EFT operators in two categories: those that can be mapped to one or more UV-complete simplified models, such as those commonly used in LHC searches so far and detailed in [23], and those for which no UV completion is available to LHC experiments, such as those outlined in Section 3.2.

### 5.2.1. EFT benchmarks with corresponding simplified models

If a simplified model can be mapped to a given EFT, then the model's high-mechator-mass limit will converge to the EFT. A study of 14 TeV benchmarks for narrow resonances with $g_q = 0.25$ and $g_\chi = 1$ (see Section 2.1.1) shows that a mediator with a mass of at least 10 TeV fully reproduces the kinematics of a contact interaction and has no remaining dependence on the presence of a resonance. A comparison of the main kinematic variables for the s-channel vector mediator model with a width of $0.1 M_{\text{med}}$ is shown in Fig. 70.<fnref n="17" />

As already observed in Section 2.1.1, varying the DM mass changes the kinematics, both in the simplified model and in the EFT case. This can be seen in Fig. 71.

Based on these studies, the Forum recommends experimental collaborations to add one grid scan point at very high mediator mass (10 TeV) to the scan, for each of the DM masses for the s-channel simplified models described in Section 2. This will allow to reproduce the results of an equivalent contact interaction as a simple extension of the existing parameter scan.

It should be checked that the high-mass mediator case for the simplified model is correctly implemented

### 5.2.2. EFT benchmarks with no corresponding simplified models

Whenever a UV completion is not available, an EFT still captures a range of possible theories beyond the simplified models that we already consider. However, in the case of the dimension-7

operators detailed in Section 3.2 we can only roughly control how well the EFT approximation holds, as described in Section 3.2.4. Despite the fact that a propagator was introduced to motivate the truncation procedure for s-channel models, the prescription from Section 5.2.1 depends upon the simplified model to derive the energy scaling that is used for the comparison with the momentum transfer. The simple fact remains that the effective coupling of the operator $-g/\Lambda^n$ - should not allow momentum flow $Q > \Lambda$ or $g > 4\pi$. Given our ignorance of the actual kinematics, the truncation procedure recommended for this purpose is the one described in Section 5.1.2, as it is independent from any UV completion details.

Because there is no UV completion, the parameter $M_{\text{cut}}$ can be treated more freely than an explicit function of $g$ and $\Lambda$. It makes sense to choose $M_{\text{cut}}$ such that we identify the transition region where the EFT stops being a good description of UV complete theories. This can be done using the ratio $R$, which is defined as the fraction of events for which $\hat{s} > M_{\text{cut}}^2$. For large values of $M_{\text{cut}}$, no events are thrown away in the truncation procedure, and $R = 1$. As $M_{\text{cut}}$ becomes smaller, eventually all events are thrown away in the truncation procedure, i.e. $R = 0$, and the EFT gives no exclusion limits for the chosen acceptance.

We propose a rough scan over $M_{\text{cut}}$, such that we find the values of $M_{\text{cut}}$ for which $R$ ranges from 0.1 to 1. The analysis can then perform a scan over several values of $M_{\text{cut}}$, and show the truncated limit for each one of them.

## 6. Evaluation of signal theoretical uncertainties

A comprehensive and careful assessment of signal theoretical uncertainties plays in general a more important role for the background estimations (especially when their evaluation is non-entirely data-driven) than it does for signal simulations. Nevertheless, also for signal samples theoretical uncertainties are relevant, and may become even dominant in certain regions of phase space.

The uncertainties on the factorization and renormalization scales are assessed by the experimental collaborations by varying the original scales of the process by factors of 0.5 and 2. The evaluation of the uncertainty on the choice of PDF follows the PDF4LHC recommendation [148] of considering the envelope of different PDF error sets, in order to account for the uncertainty on the various PDFs as well as the uncertainty on the choice of the central value PDF. The Forum has not discussed the uncertainties related to the merging of different samples, nor the uncertainty due to the choice of the modeling of the parton shower. This Chapter provides technical details on how scale and PDF uncertainties can be assessed for events generated with POWHEG and MADGRAPH5 AMC@NLO.

### 6.1. POWHEG

When using POWHEG [69–71], it is possible to study scale and PDF errors for the dark matter signals. A fast reweighting machinery is available in POWHEG-BOX that allows one to add, after each event, new weights according to different scale or PDF choices, without the need to regenerate all the events from scratch.

To enable this possibility, the variable `storeinfo_rwlock` should be set to 1 in the POWHEG input file when the events are generated for the first time.<fnref n="18" /> After each event, a line starting with

#rwgt

<footnote n="17">17 The use of a fixed width rather than the minimal width is exclusive of these plots.</footnote>

<footnote n="18">18 Notice that even if the variable is not present, by default it is set to 1.</footnote>
<figure><img src="image_94.png" /><figcaption>**Fig. 70.** Comparison of the kinematic distributions at 14 TeV between a narrow s-channel mediator and the corresponding D5 contact operator, at generator level for a jet+$\not{E}_T$ signature.</figcaption></figure>

is appended, containing the necessary information to generate extra weights. In order to obtain new weights, corresponding to different PDFs or scale choice, after an event file has been generated, a line

```
compute_rwlock 1
```

should be added in the input file along with the desired change in parameters. For instance, `renscfact` and `facscfact` allow one to study scale variations on the renormalization and factorization scales around a central value. By running the program again, a new event file will be generated, named `<OriginalName>-rwgt.lhe`, with one more line at the end of each event of the form

```
#new weight,renfact,facfact,pdf1,pdf2
```

followed by five numbers and a character string. The first of these numbers is the weight of that event with the new parameters

chosen. By running in sequence the program in the reweighting mode, several weights can be added on the same file. Two remarks are in order.

* The file with new weights is always named `<OriginalName>-rwgt.lhe` hence care has to be taken to save it as `<OriginalName>.lhe` before each iteration of the reweighting procedure.

* Due to the complexity of the environment where the program is likely to be run, it is strongly suggested as a self-consistency check that the first reweighting is done keeping the initial parameters. If the new weights are not exactly the same as the original ones, then some inconsistency must have happened, or some file was probably corrupted.

It is possible to also have weights written in the version 3 Les Houches format. To do so, in the original run, at least the token `lhrwgt_id 'ID'`
<figure><img src="image_95.png" /><figcaption>Fig. 71. Comparison of the kinematic distributions for a narrow s-channel mediator, at generator level for a jet+$\not{E}_T$ signature, for varying DM masses.</figcaption></figure>

must be present. The reweighting procedure is the same as described above, but now each new run can be tagged by using a different value for the lhrwgt_id keyword. After each event, the following lines will appear:

```
<rwgt>
<wtg id='ID'>
<wtg id='ID1'>
</rwgt>
```

A more detailed explanation of what went into the computation of every single weight can be included in the <header> section of the event file by adding/changing the line

`lhrwgt_descr 'some info'`

in the input card, before each “reweighting” run is performed. Other useful keywords to group together different weights are lhrwgt_group_name and lhrwgt_group combine.

More detailed information can be obtained by inspecting the document in /Docs/V2-paper.pdf under the common POWHEG-BOX-V2 directory.

## 6.2. The SysCalc package in MADGRAPH5_AMC@NLO

SYS CALC is a post-processing package for parton-level events as obtained from leading-order calculations in MADGRAPH5_AMC@NLO. It can associate to each event a series of weights corresponding to the evaluation of a certain class of theoretical uncertainties. The event files in input and output are compliant with the Les Houches v3 format. For NLO calculations, PDF and scale uncertainties are instead evaluated automatically by setting corresponding instructions in the run card.dat and no post-processing is needed (or possible).

The requirements of the package as inputs are:

* A systematics file (which can be generated by MadGraph 5 v. 1.6.0 or later) [75,149].

* The Pythia-PGS package (v. 2.2.0 or later) [150]. This is needed only in the case of matching scales variations.

* The availability of LHAPDF5 [151].

* A configuration file (i.e. a text file) specifying the parameters to be varied.

SYS CALC supports all leading order computations generated in MADGRAPH5_AMC@NLO including fixed-order computation and matched-merged computation performed in the MLM scheme [152]. MADGRAPH5_AMC@NLO stores additional information inside the event in order to have access to all the information required to compute the convolution of the PDFs with the matrix element for the various supported systematics.

An example configuration file is as follows:

```
# Central scale factors
scalefact:
  0.5 1 2

# Scale correlation
# Special value -1: all combination (N**2)
# Special value -2: only correlated variation
# Otherwise list of index N*fac_index + ren_index
#   index starts at 0
scalecorrelation:
  -1

# α_s emission scale factors
alpsfact:
  0.5 1 2

# matching scales
matchscale:
  30 60 120

# PDF sets and number of members (optional)
PDF:
  CT10.LHgrid 53
  MSTW2008nlo68cl.LHgrid
```

Without matching/merging, SYS CALC is able to compute the variation of renormalization and factorization scale (parameter scalefact) and the change of PDFs. The variation of the scales
can be done in a correlated and/or uncorrelated way, basically following the value of the scalecorrelation parameter which can take the following values:

* -1: to account for all $N^2$ combinations.

* -2: to account only for the correlated variations.

* A set of positive values corresponding to the following entries (assuming 0.5, 1, 2 for the scalefact entry):

$$0: \mu_F = \mu_F^{\text{orig}}/2, \mu_R = \mu_R^{\text{orig}}/2$$

$$1: \mu_F = \mu_F^{\text{orig}}/2, \mu_R = \mu_R^{\text{orig}}$$

$$2: \mu_F = \mu_F^{\text{orig}}/2, \mu_R = \mu_R^{\text{orig}} * 2$$

$$3: \mu_F = \mu_F^{\text{orig}}, \mu_R = \mu_R^{\text{orig}}/2$$

$$4: \mu_F = \mu_F^{\text{orig}}, \mu_R = \mu_R^{\text{orig}}$$

$$5: \mu_F = \mu_F^{\text{orig}}, \mu_R = \mu_R^{\text{orig}} * 2$$

$$6: \mu_F = \mu_F^{\text{orig}} * 2, \mu_R = \mu_R^{\text{orig}}/2$$

$$7: \mu_F = \mu_F^{\text{orig}} * 2, \mu_R = \mu_R^{\text{orig}}$$

$$8: \mu_F = \mu_F^{\text{orig}} * 2, \mu_R = \mu_R^{\text{orig}} * 2$$

Without correlation, the weight associated to the renormalization scale is the following:

$$\mathcal{W}_{\text{new}}^{\mu_R} = \frac{\alpha_S^N (\Delta * \mu_R)}{\alpha_S^N (\mu_R)} * \mathcal{W}_{\text{orig}}, \quad (49)$$

where $\Delta$ is the scale variation considered, $\mathcal{W}_{\text{orig}}$ and $\mathcal{W}_{\text{new}}$ are respectively the original/new weights associated to the event. $N$ is the power in the strong coupling for the associated event (interference is not taken account on an event by event basis). The weight associated to the scaling of the factorization scale is:

$$\mathcal{W}_{\text{new}}^{\mu_F} = \frac{f_{1,\text{orig}}(x_1, \Delta * \mu_F) * f_{2,\text{orig}}(x_2, \Delta * \mu_F)}{f_{1,\text{orig}}(x_1, \mu_F) * f_{2,\text{orig}}(x_2, \mu_F)} * \mathcal{W}_{\text{orig}}, \quad (50)$$

where $f_{i,\text{orig}}$ are the probabilities from the original PDF set associated to the incoming partons, which hold a proton momentum fraction $x_1$ and $x_2$ for the first and second beam respectively.

The variations for the PDF are given by the corresponding weights associated to the new PDF sets:

$$\mathcal{W}_{\text{new}}^{\text{PDF}} = \frac{f_{1,\text{new}}(x_1, \mu_F) * f_{2,\text{new}}(x_2, \mu_F)}{f_{1,\text{orig}}(x_1, \mu_F) * f_{2,\text{orig}}(x_2, \mu_F)} * \mathcal{W}_{\text{orig}}, \quad (51)$$

where $f_{i,\text{new}}$ is the new PDF probability associated to parton $i$.

In presence of matching, MADGRAPH5_AMC@NLO associates one history of radiation (initial and/or final state radiation) obtained by a $k_T$ clustering algorithm, and calculates $\alpha_s$ at each vertex of the history to a scale given by the aforementioned clustering algorithm. Furthermore, MADGRAPH5_AMC@NLO reweights the PDF in a fashion similar to what a parton shower would do. SYS CALC can perform the associated re-weighting (parameter alpsfact) by dividing and multiplying by the associated factor.

For each step in the history of the radiation (associated to a scale $\mu_i = k_{T,i}$), this corresponds to the following expression for a Final State Radiation (FSR):

$$\mathcal{W}_{\text{new}}^{\text{FSR}} = \frac{\alpha_s (\Delta * \mu_i)}{\alpha_s (\mu_i)} * \mathcal{W}_{\text{orig}}, \quad (52)$$

and to the following expression for Initial State Radiation (ISR), associated to a scale $\mu_i$ and fraction of energy $x_i$:

$$\mathcal{W}_{\text{new}}^{\text{ISR}} = \frac{\alpha_s (\Delta * \mu_i)}{\alpha_s (\mu_i)} \frac{f_a(x_i, \Delta * \mu_i)}{f_b(x_i, \Delta * \mu_i)} * \mathcal{W}_{\text{orig}}, \quad (53)$$

where $\mu_{i+1}$ is the scale of the next step in the (initial state) history of radiation.

SYS CALC can include the weight associated to different merging scales in the MLM matching/merging mechanism (for output of the pythia6 package or pythia-pgs package).

In that case, the parton shower does not veto any event according to the MLM algorithm, although in the output file the scale of the first emission is retained. Having this information, SYS CALC can test each value of the specified matching scales under the matchscale parameter block. SYS CALC will then test for each of the values specified in the parameter matchscale if the event passes the MLM criteria or not. If it does not, then a zero weight is associated to the event, while if it does, then a weight 1 is kept. As a reminder, those weights are the equivalent of having a (approximate) Sudakov form-factor and removing at the same time the double counting between the events belonging to different multiplicities.

Finally, we give an example of the SYS CALC output which follows the LHEF v3 format. The following block appears in the header of the output file:

```
header
<initrwgt>
<weightgroup type="Central scale variation" combine="envelope">
<weight id="1"> mur=0.5 muf=0.5 </weight>
<weight id="2"> mur=1 muf=0.5 </weight>
<weight id="3"> mur=2 muf=0.5 </weight>
<weight id="4"> mur=0.5 muf=1 </weight>
<weight id="5"> mur=1 muf=1 </weight>
<weight id="6"> mur=2 muf=1 </weight>
<weight id="7"> mur=0.5 muf=2 </weight>
<weight id="8"> mur=1 muf=2 </weight>
<weight id="9"> mur=2 muf=2 </weight>
</weightgroup>
<weightgroup type="Emission scale variation" combine="envelope">
<weight id="10"> alpsfact=0.5 </weight>
<weight id="11"> alpsfact=1 </weight>
<weight id="12"> alpsfact=2 </weight>
</weightgroup>
<weightgroup type="CT10nlo.LHgrid" combine="hessian">
<weight id="13">Member 0</weight>
<weight id="14">Member 1</weight>
<weight id="15">Member 2</weight>
<weight id="16">Member 3</weight>
...
<weight id="65">Member 52</weight>
</weightgroup>
</initrwgt>
</header>
```

For each event, the weights are then written as follows:

```
<rwgt>
<wtgt id="1">83214.7</wtgt>
<wtgt id="2">61460</wtgt>
<wtgt id="3">47241.9</wtgt>
<wtgt id="4">101374</wtgt>
...
<wtgt id="64">34893.5</wtgt>
<wtgt id="65">41277</wtgt>
</rwgt>
```

## 7. Conclusions

The ATLAS/CMS Dark Matter Forum concluded its work in June 2015. Its mandate was focused on identifying a prioritized, compact set of simplified model benchmarks to be used for the design of the early Run-2 LHC searches for $\ell_T + X$ final states. Its participants included many of the experimenters from both collaborations that are involved in these searches, as well as many of the theorists working actively on these models. This report has documented this basis set of models, as well as studies of the kinematically-distinct regions of the parameter space of the models, to aid the design of the searches. Table 9 summarizes the state of the art of the calculations, event generators, and tools that are available to the two LHC collaborations to simulate
<caption>**Table 9**</caption>

Summary table for available benchmark models considered within the works of this Forum. The results in this document have been obtained with the implementations in bold.

<table><thead><tr><th colspan="4">Benchmark models for ATLAS and CMS Run-2 DM searches</th></tr><tr><th colspan="4">Vector/axial vector mediator, s-channel (Section 2.1)</th></tr><tr><th>Signature</th><th>State of the art calculation and tools</th><th>Implementation</th><th>References</th></tr></thead><tbody><tr><td>jet + E<sub>T</sub></td><td>NLO + PS (POWHEG, SVN r3059)<br>NLO+PS (DMsimp UFO + MADGRAPH5_AMC@NLO v2.3.0)<br>NLO (MCFM v7.0)</td><td>[115,116]<br>[113]<br>Upon request</td><td>[43,61,69-71]<br>[75-77]<br>[41,114]</td></tr><tr><td>W/Z/γ + E<sub>T</sub></td><td>LO + PS (UFO + MADGRAPH5_AMC@NLO v2.2.3)<br>NLO+PS (DMsimp UFO + MADGRAPH5_AMC@NLO v2.3.0)</td><td>[126]<br>[113]</td><td>[75-77]<br>[75-77]</td></tr><tr><th colspan="4">Scalar/pseudoscalar mediator, s-channel (Section 2.2)</th></tr><tr><th>Signature</th><th>State of the art calculation and tools</th><th>Implementation</th><th>References</th></tr><tr><td>jet + E<sub>T</sub></td><td>LO + PS, top loop (POWHEG, r3059)<br>LO+PS, top loop (DMsimp UFO + MADGRAPH5_AMC@NLO v2.3.0)<br>LO, top loop (MCFM v7.0)</td><td>[117,118]<br>[113]<br>Upon request</td><td>[43,61,69-71]<br>[75-77,153]<br>[41,114]</td></tr><tr><td>W/Z/γ + E<sub>T</sub></td><td>LO + PS (UFO + MADGRAPH5_AMC@NLO v2.2.3)</td><td>[127]</td><td>[75-77]</td></tr><tr><td>t<span style="text-decoration: overline;">t</span>, b<span style="text-decoration: overline;">b</span> + E<sub>T</sub></td><td>LO + PS (UFO + MADGRAPH5_AMC@NLO v2.2.3)<br>NLO+PS (DMsimp UFO + MADGRAPH5_AMC@NLO v2.3.0)</td><td>[113]</td><td>[75-77]<br>[75-77]</td></tr><tr><th colspan="4">Scalar mediator, t-channel (Section 2.3)</th></tr><tr><th>Signature</th><th>State of the art calculation and tools</th><th>Implementation</th><th>References</th></tr><tr><td>jet(s) + E<sub>T</sub> (2-quark gens.)</td><td>LO+PS (UFO + MADGRAPH5_AMC@NLO v2.2.3)</td><td>[124]</td><td>[75-78]</td></tr><tr><td>jet(s) + E<sub>T</sub> (3-quark gens.)</td><td>LO + PS (UFO + MADGRAPH5_AMC@NLO v2.2.3)</td><td>[125]</td><td>[46,75-77]</td></tr><tr><td>W/Z/γ + E<sub>T</sub></td><td>LO + PS (UFO + MADGRAPH5_AMC@NLO v2.2.3)</td><td>TBC</td><td>[46,75-77]</td></tr><tr><td>b + E<sub>T</sub></td><td>LO + PS (UFO + MADGRAPH5_AMC@NLO v2.2.3)</td><td>[128]</td><td>[75-77,87,88]</td></tr><tr><th colspan="4">Specific simplified models with EW bosons (Section 3.1)</th></tr><tr><th>Signature and model</th><th>State of the art calculation and tools</th><th>Implementation</th><th>References</th></tr><tr><td>Higgs + E<sub>T</sub>, vector med.</td><td>LO + PS (UFO + MADGRAPH5_AMC@NLO v2.2.3)</td><td>[137]</td><td>[48,75-77,97]</td></tr><tr><td>Higgs + E<sub>T</sub>, scalar med.</td><td>LO + PS (UFO + MADGRAPH5_AMC@NLO v2.2.3)</td><td>[137]</td><td>[48,75-77,97]</td></tr><tr><td>Higgs + E<sub>T</sub>, 2HDM</td><td>LO + PS (UFO + MADGRAPH5_AMC@NLO v2.2.3)</td><td>[138]</td><td>[75-77,97]</td></tr><tr><th colspan="4">Contact interaction operators with EW bosons (Section 3.1)</th></tr><tr><th>Signature and model</th><th>State of the art calculation and tools</th><th>Implementation</th><th>References</th></tr><tr><td>W/Z/γ + E<sub>T</sub>, dim-7</td><td>LO + PS (UFO + MADGRAPH5_AMC@NLO v2.2.3)</td><td>[141]</td><td>[45,75-77,97-99]</td></tr><tr><td>Higgs + E<sub>T</sub>, dim-4/dim-5</td><td>LO + PS (UFO + MADGRAPH5_AMC@NLO v2.2.3)</td><td>[154]</td><td>[47,48,75-77,97]</td></tr><tr><td>Higgs + E<sub>T</sub>, dim-8</td><td>LO + PS (UFO + MADGRAPH5_AMC@NLO v2.2.3)</td><td>[137]</td><td>[47,48,75-77,97]</td></tr></tbody></table>

these models at the start of Run-2. It also describes some that are known to be under development as the report was finalized.

This document primarily presents studies related to simplified models. The presentation of results for EFT benchmark models is also discussed. The studies contained in this report are meant to highlight the use of EFTs as a benchmark that is complementary to simplified models, and to demonstrate how that collider results could be presented a function of the fraction of events that are valid within the contact interaction approximation.

A number of points remain to be developed beyond the scope of this Forum, in order to fully benefit from LHC searches in the global quest for Dark Matter. First and foremost, to accommodate the urgent need of a basis set of simplified models, this work has made many grounding assumptions, as stated in the introduction. Departures from these assumptions have not been fully explored. As a consequence, the list of models and implementations employed by the ATLAS and CMS collaborations for early LHC Run-2 searches is not meant to exhaust the range of possibilities for mediating processes, let alone cover all plausible models of collider dark matter production. Rather, it is hoped that others will continue the systematic exploration of the most generic possibilities for collider dark matter production, building upon the framework used in this report just as this report has relied heavily on the work of many others. This also applies to models that exist in literature but do not have an implementation yet: we hope that this work will further encourage the theory and generator community to improve the implementation of new models as well as the precision of the calculations of existing ones. The role of constraints on the mediator particles from

direct past and present collider searches should also be developed further.

Furthermore, we see the need for broader discussion on the comparison of experimental results amongst collider and non-collider searches for particle dark matter. This point will have to be addressed before the presentation of Run-2 results: The uncertainties in the comparisons between experiments should be discussed and conveyed, so that the different results can be placed in their correct context, and so we can collectively build a fair and comprehensive picture of our understanding of particle dark matter.

## Acknowledgments

The authors would like to thank Daniel Whiteson for helping in the review of this document. This research was supported by the Munich Institute for Astro- and Particle Physics (MI-APP) of the DFG cluster of excellence “Origin and Structure of the Universe”. The authors would like to express a special thanks to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and support. P. Pani wishes to thank the support of the Computing Infrastructure of Nikhef. In addition we thank CERN and acknowledge the support of: DOE and NSF (USA), including grant DE-SC0011726 for the preparation of this manuscript; SER, SNSF and Cantons of Bern and Geneva, Switzerland; MINECO, Spain; European Union, where the preparation of this manuscript is part of a project that has received funding from the European Research Council under the European Union's Horizon 2020 research and innovation program (grant agreement 679305); IN2P3-CNRS, CEA-DSM/IRFU, France; NSERC,
NRC, and CFI, Canada; INFN, Italy; FNRS and FWO, Belgium; STFC, United Kingdom; ARC, Australia; GRICES and FCT, Portugal; SRC, Sweden; BMBF, DFG, HGF, MPG and AvH Foundation, Germany; Austrian Academy of Sciences, Austria; MOE and UM, Malaysia; National Research Foundation and Ministry of Education, Science and Technology, Korea; MEXT and JSPS, Japan; FOM and NWO, Netherlands; CNPq and FAPESP, Brazil; Atomic Energy Commission, Pakistan; DNRF, DNSRC and Lundbeck Foundation, Denmark; Thailand Center of Excellence in Physics, Institute for the Promotion of Teaching Science and Technology, Special Task Force for Activating Research and National Science and Technology Development Agency, Thailand; MOST, Taiwan.

## Appendix A. Additional models for dark matter searches

### A.1. Models with a single top-quark + $\ell^\pm T$

Many different theories predict final states with a single top and associated missing transverse momentum (monotop), some of them including dark matter candidates. A simplified model encompassing the processes leading to this phenomenology is described in Refs. [155–157], and is adopted as one of the benchmarks for Run 2 LHC searches.

The simplified model is constructed by imposing that the model Lagrangian respects the electroweak $SU(2)_L \times U(1)_Y$ gauge symmetry and by requiring minimality in terms of new states to supplement to the Standard Model fields. As a result, two monotop production mechanisms are possible. In the first case, the monotop system is constituted by an invisible (or long-lived with respect to detector distances) fermion $\chi$ and a top quark. It is produced as shown in the diagram of A.72(a) where a colored resonance $\varphi$ lying in the triplet representation of $SU(3)_c$ decays into a top quark and a $\chi$ particle. In the second production mode, the monotop state is made of a top quark and a vector state $V$ connected to a hidden sector so that it could decay invisibly into, e.g., a pair of dark matter particles as studied in [157]. The production proceeds via flavor-changing neutral interactions of the top quark with a quark of the first or second generation and the invisible $V$ boson (see the diagrams of A.72(b) and (c)).

**Resonant production.** In this case, a colored 2/3-charged scalar ($\varphi$) is produced and decays into a top quark and a spin-1/2 invisible particle, $\chi$. The dynamics of the new sector is described by the following Lagrangian:

$$ \mathcal{L} = \left[ \varphi \bar{d}^c \left[ a_{SR}^q + b_{SR}^q \gamma_5 \right] d + \varphi \bar{u} \left[ a_{SR}^{1/2} + b_{SR}^{1/2} \gamma_5 \right] \chi + \text{h.c.} \right], \quad (A.1) $$

where $u(d)$ stands for any up-type (down-type) quark, the notation $SR$ refers to the monotop production mechanism via a scalar resonance and all flavor and color indices are understood for clarity.

In the notation of [156], the couplings of the new colored fields to down-type quarks are embedded into the $3 \times 3$ antisymmetric matrices $a_{SR}^q$ (scalar couplings) and $b_{SR}^q$ (pseudoscalar couplings) while those to the new fermion $\chi$ and one single up-type quark are given by the three-component vectors $a_{SR}^{1/2}$ and $b_{SR}^{1/2}$ in flavor space.

Under the form of Eq. (A.1), the Lagrangian is the one introduced in the original monotop search proposal [155]. It has been used by the CMS collaboration for Run I analyses after neglecting all pseudoscalar components of the couplings and adding the vector resonance case for which minimality requirements are difficult to accommodate [18]. In contrast, the study of Ref. [157] has imposed electroweak gauge invariance and required minimality.

This enforces all new couplings to be right-handed so that

$$ a_{SR}^{1/2} = b_{SR}^{1/2} = \frac{1}{2} y_s^* \quad \text{and} \quad a_{SR}^q = b_{SR}^q = \frac{1}{2} \lambda_s, \qquad (A.2) $$

where the objects $y_s$ and $\lambda_s$ are a tridimensional vector and a $3 \times 3$ matrix in flavor space respectively. This class of scenarios is the one that has been adopted by the ATLAS collaboration for its Run I monotop searches [12] and will be considered by both collaborations for Run II analyses.

The resulting model can be likened to the MSSM with an R-parity violating of a top squark to the Standard Model down-type quarks and an R-parity conserving interaction of a top quark and a top-squark to a neutralino.

**Non-resonant production.** For non-resonant monotop production, the monotop state is produced via flavor-changing neutral interactions of the top quark, a lighter up-type quark and a new invisible vector particle $V$. This is the only case considered, as having a new scalar would involve in particular a mixing with the SM Higgs boson and therefore a larger number of free parameters. The Lagrangian describing the dynamics of this non-resonant monotop production case is:

$$ \mathcal{L} = \left[ V_\mu \bar{u} \gamma^\mu \left[ a_{FC}^1 + b_{FC}^1 \gamma_5 \right] u + \text{h.c.} \right], \quad (A.3) $$

where the flavor and color indices are again understood for clarity. The strength of the interactions among these two states and a pair of up-type quarks is modeled via two $3 \times 3$ matrices in flavor space $a_{FC}^1$ for the vector couplings and $b_{FC}^1$ for the axial vector couplings, the FC subscript referring to the flavor-changing neutral monotop production mode and the (1) superscript to the vectorial nature of the invisible particle.

As for the resonant case, the Lagrangian of Eq. (A.3) is the one that has been used by CMS after reintroducing the scalar option for the invisible state and neglecting all pseudoscalar interactions [18]. As already mentioned, a simplified setup motivated by gauge invariance and minimality has been preferred so that, as shown in Ref. [157], we impose all interactions to involve right-handed quarks only,<fnref n="19" />

$$ a_{FC}^1 = b_{FC}^1 = \frac{1}{2} a_R \qquad (A.4) $$

where $a_R$ denotes a $3 \times 3$ matrix in flavor space. This implies the vector field to be an $SU(2)_L$ singlet.

**Model parameters and assumptions.** The models considered as benchmarks for the first LHC searches contain further assumptions in terms of the flavor structure of the model with respect to the Lagrangians of the previous subsection. In order to have an observable monotop signature at the LHC, the Lagrangians introduced above must include not too small couplings of the new particles to first and second generation quarks. For simplicity, we assumed that only channels enhanced by parton density effects will be considered, so that we fix

$$ \begin{gathered} (a_R)_{13} = (a_R)_{31} = a, \\ (\lambda_s)_{12} = -(\lambda_s)_{21} = \lambda \quad \text{and} \quad (y_s)_3 = y, \end{gathered} \qquad (A.5) $$

all other elements of the matrices and vectors above being set to zero.

**Implementation.** In order to allow one for the Monte Carlo simulation of events relevant for the monotop production cases

<footnote n="19">¹⁹ Ref. [158], which became public only after the conclusion of this Forum, introduces both left- and right-handed couplings.</footnote>
<figure><img src="image_96.png" /><figcaption>Fig. A.72. Feynman diagrams of leading order processes leading to monotop events: production of a colored scalar resonance $\varphi$ decaying into a top quark and a spin-1/2 fermion $\chi$ (a), s- (b) and t-channel (c) non resonant production of a top quark in association with a spin-1 boson V decaying invisibly.</figcaption></figure>

described above, we consider the Lagrangian

$$ \mathcal{L} = \left[ aV_{\mu}\bar{u}\gamma^{\mu}P_{R}t + \lambda\varphi\bar{d}^{c}P_{R}s + y\varphi\bar{\chi}P_{R}t + \text{h.c.} \right], \quad (A.6) $$

where $P_R$ stands for the right-handed chirality projector and the new physics couplings are defined by the three parameters $a, \lambda$ and $y$. We additionally include a coupling of the invisible vector boson V to a dark sector (represented by a fermion $\psi$) whose strength can be controlled through a parameter $g_{DM}$,

$$ \mathcal{L} = g_{DM}V_{\mu}\bar{\psi}\gamma^{\mu}\psi. \quad (A.7) $$

This ensures the option to make the V-boson effectively invisible by tuning $g_{DM}$ respectively to $a$. We implement the entire model in the FEYNRULES package [76] so that the model can be exported to a UFO library [77] to be linked to MADGRAPH5_AMC@NLO [75] for event generation, following the approach outlined in [159].

## A.2. Parameter scan

Under all the assumptions of the previous sections, the parameter space of the resonant model is defined by four quantities, namely the mass of the new scalar field $\varphi$, the mass of the invisible fermion $\chi$ and the strengths of the interactions of the scalar resonance with the monotop system y and with down-type quarks $\lambda$. One of both coupling parameters could however be traded with the width of the resonance.

The parameter space of the non-resonant model is defined by two parameters, namely the mass of the invisible state V and its flavor-changing neutral coupling to the up-type quarks $a_R$.

In the case of the non-resonant model, the invisible vector is connected to a hidden sector that could be, in its simplest form, parameterized by a new fermion [157]. This has effects on the width of the invisible V state.

A consensus between the ATLAS and CMS collaborations has been reached in the case of non-resonant monotop production. The results have been described above. In contrast, discussions in the context of resonant monotop production are still on-going. The related parameter space contains four parameters and must thus be further simplified for practical purposes. Several options are possible and a choice necessitates additional studies that will be achieved in a near future.

It has been verified that the kinematics do not depend on the width of the invisible state in the case where this width is at most 10% of the V-mass. This is illustrated in Fig. A.73, where we show the transverse-momentum spectra of the V-boson when it decays into a top-up final state and for different V-boson masses. The results are independent of the visible or invisible decay modes as we are only concerned with the kinematic properties of the invisible state.

## A.3. Single top model implementation

Card files for MADGRAPH5_AMC@NLO are provided on the Forum Git repository [160] and correspond to the Lagrangian that has been implemented in FEYNRULES. Each coupling constant of the model can be set via the block COUPX of the parameter card. Its entries 1, 2 and 3 respectively correspond to the monotop-relevant parameters $a, \lambda$ and $y$, while the width (and in particular the invisible partial width) of the V-boson can be tuned via the $g_{DM}$ parameter to given in the entry 10 of the COUPX block.

The masses of the particles are set in the MASS block of the parameter card, the PDG codes of the new states being 32 (the vector state V), 1000006 (the $\varphi$ colored resonance), 1000022 (the invisible fermion $\chi$) and 1000023 (the fermion $\psi$ connecting the V state to the dark sector). The width of the new vector has to be computed from all open tree-level decays (after fixing $g_{DM}$ to a large value and setting the relevant entry to Auto in the DECAY
<figure><img src="image_97.png" /><figcaption>**Fig. A.73.** Distributions of the transverse momentum of the V boson in the case of the process $pp \to tV \to t(t\bar{u} + \text{c.c.})$. We have imposed that the V-boson is produced on-shell and have chosen its mass to be $m_V = 200, 600$ and $1000$ GeV (left, central and right panels). We have considered three possible cases for the total width of the V-boson, which has been fixed to 0.61%, 0.1% and 10% of the mass.</figcaption></figure>

block of the parameter card), while the way to calculate the width of the resonance $\phi$ is under discussion by both the ATLAS and CMS collaborations. The chi and psi fermions are taken stable so that their width vanishes.

### A.4. Further W+$\bar{E}_T$ models with possible cross-section enhancements

As pointed out in Ref. [67], the mono-W signature can probe the iso-spin violating interactions of dark matter with quarks. The relevant operator after the electroweak symmetry breaking is

$$ \frac{1}{\Lambda^2} \bar{\chi} \gamma_\mu \chi (\bar{u}_L \gamma^\mu u_L + \xi \bar{d}_L \gamma^\mu d_L). \qquad (A.8) $$

Here, we only keep the left-handed quarks because the right-handed quarks do not radiate a W-gauge boson from the weak interaction. As the LHC constrains the cutoff to higher values, it is also important to know the corresponding operators before the electroweak symmetry. At the dimension-six level, the following operator

$$ \frac{c_6}{\Lambda^2} \bar{\chi} \gamma_\mu \chi \bar{Q}_L \gamma^\mu Q_L \qquad (A.9) $$

conserves iso-spin and provides us $\xi = 1$ [67]. At the dimension-eight level, new operators appear to induce iso-spin violation and can be

$$ \frac{c_8^d}{\Lambda^4} \bar{\chi} \gamma_\mu \chi (H \bar{Q}_L) \gamma^\mu (Q_L H^\dagger) + \frac{c_8^u}{\Lambda^4} \bar{\chi} \gamma_\mu \chi (\bar{H} \bar{Q}_L) \gamma^\mu (Q_L \bar{H}^\dagger). \qquad (A.10) $$

After inputting the vacuum expectation value of the Higgs field, we have

$$ \xi = \frac{c_6 + c_8^d v_{EW}^2 / 2\Lambda^2}{c_6 + c_8^u v_{EW}^2 / 2\Lambda^2}. \qquad (A.11) $$

For a nonzero $c_6$ and $v_{EW} \ll \Lambda$, the iso-spin violation effects are suppressed. On the other hand, the values of $c_6$, $c_8^d$ and $c_8^u$ depend on the UV-models.

There is one possible UV-model to obtain a zero value for $c_6$ and non-zero values for $c_8^d$ and $c_8^u$. One can have the dark matter and the SM Higgs field charged under a new $U(1)'$ symmetry. There is a small mass mixing between SM Z-boson and the new Z' with a mixing angle of $\phi(v_{EW}^2/M_Z^2)$. After integrating out $Z'$, one has different effective dark matter couplings to $u_L$ and $d_L$ fields, which are proportional to their couplings to the Z boson. For this model, we have $c_6 = 0$ and

$$ \xi = \frac{-\frac{1}{2} + \frac{1}{3} \sin^2 \theta_W}{\frac{1}{2} - \frac{2}{3} \sin^2 \theta_W} \approx -2.7 \qquad (A.12) $$

and order of unity.

### A.5. Simplified model corresponding to dimension-5 EFT operator

As an example of a simplified model corresponding to the dimension-5 EFT operator described in Section 3.2, we consider a Higgs portal with a scalar mediator. Models of this kind are among the most concise versions of simplified models that produce couplings of Dark Matter to pairs of gauge-bosons. Scalar fields may couple directly to pairs of electroweak gauge bosons, but must carry part of the electroweak vacuum expectation value. One may thus consider a simple model where Dark Matter couples to a scalar singlet mediator, which mixes with the fields in the Higgs sector.

$$ L \subset \frac{1}{2} m_s S^2 + \lambda S^2 |H|^2 + \lambda' S |H|^2 + y S \chi \bar{\chi} \qquad (A.13) $$

where H is a field in the Higgs sector that contains part of the electroweak vacuum expectation value, S is a heavy scalar singlet and $\chi$ is a Dark Matter field. There is then an s-channel diagram where DM pairs couple to the singlet field S, which then mixes with a Higgs-sector field, and couples to W and Z bosons. This diagram contains 2 insertions of EW symmetry breaking fields, corresponding in form to the effective dimension-5 operator in Section 3.2.1.

### A.6. Inert two-Higgs Doublet Model (IDM)

For most of the simplified models included in this report, the mass of the mediator and couplings/width are non-trivial parameters of the model. In these scenarios, we remain agnostic about the theory behind the dark matter sector and try to parameterize it in simple terms.

We have not addressed how to extend the simplified models to realistic and viable models which are consistent with the symmetries of the Standard Model. Simplified models often violate gauge invariance which is a crucial principle for building a consistent BSM model which incorporates SM together with new physics. For example, with a new heavy gauge vector boson mediating DM interactions, one needs not just the dark matter and its mediator, but also a mechanism which provides mass to this mediator in a gauge invariant way.

Considering both the simplified model and other elements necessary for a consistent theory is a next logical step. The authors of [161] term these Minimal Consistent Dark Matter (MCDM) models. MCDM models are at the same time still toy models that can be easily incorporated into a bigger BSM model and explored via complementary constraints from collider and direct/indirect DM search experiments as well as relic density constraints.
<figure><img src="image_98.png" /><figcaption>**Fig. A.74.** Feynman diagrams for $gg \to h_1h_1 + g$ process contributing to mono-jet signature. Source: Adapted from [161].</figcaption></figure>

<figure><img src="image_99.png" /><figcaption>**Fig. A.75.** Feynman diagrams for $q\bar{q} \to h_1h_2 + g$ ($gq \to h_1h_2 + q$) process contributing to mono-jet signature. Source: Adapted from [161].</figcaption></figure>

<figure><img src="image_100.png" /><figcaption>**Fig. A.76.** Feynman diagrams for $q\bar{q} \to h_1h_1 + Z$ process contributing to mono-Z signature. Source: Adapted from [161].</figcaption></figure>

The idea of an inert Two-Higgs Doublet Model (IDM) was introduced more than 30 years ago in Ref [162]. The IDM was first proposed as a Dark Matter model in Ref. [163] and its phenomenology further studied in Refs. [14,161,164–173]. It is an extension of the SM with a second scalar doublet $\phi_2$ with no direct coupling to fermions. This doublet has a discrete $Z_2$ symmetry, under which $\phi_2$ is odd and all the other fields are even. The Lagrangian of the odd sector is,

$$ \mathcal{L} = \frac{1}{2}(D_{\mu}\phi_2)^2 - V(\phi_1, \phi_2) \qquad (A.14) $$

with the potential $V$ containing mass terms and $\phi_1 - \phi_2$ interactions:

$$ \begin{aligned} V = & -m_1^2(\phi_1^\dagger\phi_1) - m_2^2(\phi_2^\dagger\phi_2) + \lambda_1(\phi_1^\dagger\phi_1)^2 + \lambda_2(\phi_2^\dagger\phi_2)^2 \\ & + \lambda_3(\phi_2^\dagger\phi_2)(\phi_1^\dagger\phi_1) + \lambda_4(\phi_2^\dagger\phi_1)(\phi_1^\dagger\phi_2) \\ & + \frac{\lambda_5}{2}[(\phi_1^\dagger\phi_2)^2 + (\phi_2^\dagger\phi_1)^2], \end{aligned} \qquad (A.15) $$

where $\phi_1$ and $\phi_2$ are SM and inert Higgs doublets respectively carrying the same hypercharge. These doublets can be parameterized as

$$ \phi_1 = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ v + H \end{pmatrix} \qquad \phi_2 = \frac{1}{\sqrt{2}} \begin{pmatrix} \sqrt{2}h^+ \\ h_1 + ih_2 \end{pmatrix} \qquad (A.16) $$

In addition to the SM, the IDM introduces four more degrees of freedom coming from the inert doublet in the form of a $Z_2$-odd charged scalar $h^\pm$ and two neutral $Z_2$-odd scalars $h_1$ and $h_2$. The lightest neutral scalar, $h_1$ is identified as the dark matter candidate. Aspects of the IDM collider phenomenology have been studied in [161,163,173–184]. Its LHC signatures include dileptons [179,184], trileptons [180] and multileptons [181] along with

missing transverse energy, modifications of the Higgs branching ratios [173,182,183], as well as $E_T$-jet, Z, and Higgs and $E_T$-VBF signals (see Figs. A.74–A.79).

Based on the various LHC search channels, DM phenomenology issues and theoretical considerations, numerous works have proposed benchmark scenarios for the IDM, see e.g. [173,181] while a FeynRules implementation (including MadGraph, CalcHEP and micrOMEGAs model files) was provided in [181]. An updated analysis of the parameter space has recently been performed in Ref. [161].

The authors suggested to study mono-X signatures that are relevant to model-independent collider DM searches, and evaluated their rates presented below. They have implemented and cross-checked the IDM model into CalcHEP and micrOMEGAs, with an implementation publicly available on the HEPMDB database, including loop-induced HHG and $\gamma\gamma H$ models. They propose an additional set of benchmark points, mostly inspired by mono-X and VBF searches (Table A.10). Though the overall parameter space of IDM is 5-dimensional, once all relevant constraints are applied the parameter space relevant to a specific LHC signature typically reduces to 1–2 dimensional. In the mono-jet case, one can use two separate simplified models, a $gg \to h_1h_1 + g$ process (via Higgs mediator) and a $qq \to h_1h_2 + g$ ($gq \to h_1h_2 + q$) process (through a Z-boson mediator) to capture the physics relevant to the search. The cross sections for the various mono-X and VBF signatures produced by this model are displayed in Fig. A.80.

## Appendix B. Presentation of experimental results for reinterpretation
<figure><img src="image_101.png" /><figcaption>**Fig. A.77.** Feynman diagrams for $gg \to h_1h_1 + H$ process contributing to mono-Higgs signature.
Source: Adapted from [161].</figcaption></figure>

<figure><img src="image_102.png" /><figcaption>**Fig. A.78.** Feynman diagrams for $q\bar{q} \to h_1h_2 + H$ process contributing to mono-Higgs signature.
Source: Adapted from [161].</figcaption></figure>

<figure><img src="image_103.png" /><figcaption>**Fig. A.79.** Diagrams for $qq \to qqh_1h_1$ DM production in vector boson fusion process.
Source: Adapted from [161].</figcaption></figure>

<caption>**Table A.10**</caption>

Five benchmarks for IDM in ($M_{h_1}$, $M_{h_2}$, $M_{h_\pm}$, $\lambda_2$, $\lambda_{345}$) parameter space. We also present the corresponding relic density ($\Omega h^2$), the spin-independent cross section for DM scattering on the proton ($\sigma_{SI}$), and the LHC cross section at 13 TeV for mono-jet process $pp \to h_1, h_1 + \text{jet}$ for $p_T^{\text{jet}} > 100$ GeV cut ($\sigma_{LHC}$).

<table><thead><tr><th>BM</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><td>$M_{h_1}$ (GeV)</td><td>48</td><td>53</td><td>70</td><td>82</td><td>120</td></tr><tr><td>$M_{h_2}$ (GeV)</td><td>55</td><td>189</td><td>77</td><td>89</td><td>140</td></tr><tr><td>$M_{h_\pm}$ (GeV)</td><td>130</td><td>182</td><td>200</td><td>150</td><td>200</td></tr><tr><td>$\lambda_2$</td><td>0.8</td><td>1.0</td><td>1.1</td><td>0.9</td><td>1.0</td></tr><tr><td>$\lambda_{345}$</td><td>-0.010</td><td>-0.024</td><td>+0.022</td><td>-0.090</td><td>-0.100</td></tr><tr><td>$\Omega h^2$</td><td>3.4 &times; 10<sup>-2</sup></td><td>8.1 &times; 10<sup>-2</sup></td><td>9.63 &times; 10<sup>-2</sup></td><td>1.5 &times; 10<sup>-2</sup></td><td>2.1 &times; 10<sup>-3</sup></td></tr><tr><td>$\sigma_{SI}$ (pb)</td><td>2.3 &times; 10<sup>-10</sup></td><td>7.9 &times; 10<sup>-10</sup></td><td>5.1 &times; 10<sup>-10</sup></td><td>4.5 &times; 10<sup>-10</sup></td><td>2.6 &times; 10<sup>-9</sup></td></tr><tr><td>$\sigma_{LHC}$ (fb)</td><td>1.7 &times; 10<sup>2</sup></td><td>7.7 &times; 10<sup>2</sup></td><td>4.3 &times; 10<sup>-2</sup></td><td>1.2 &times; 10<sup>-1</sup></td><td>2.3 &times; 10<sup>-2</sup></td></tr></tbody></table>

When collider searches present results with the recommended benchmarks, we suggest the following:

* Provide limits in collider language, on fundamental parameters of the interaction: the couplings and masses of particles in simplified model.

* Translate limits to non-collider language, for a range of assumptions, in order to convey a rough idea of the range of possibilities. The details of this point are left for work beyond the scope of this Forum.

* Provide all necessary material for theorists to reinterpret simplified model results as building blocks for more complete models (e.g. signal cutflows, acceptances, etc.). This point is detailed further in this appendix.

* Provide model-independent results in terms of limits on cross-section times efficiency times acceptance of new phenomena for all cases, but especially when EFTs are employed as benchmarks. This recommendation has been issued before: see Ref. [185] for detailed suggestions.
<figure><img src="image_104.png" /><figcaption>Fig. A.80. LHC cross section at 13 TeV for various signatures, from [161].</figcaption></figure>

* Provide easily usable and clearly labeled results in a digitized format, e.g. [186] entries, ROOT histograms and macros or tables available on analysis public pages.

This appendix describes further considerations for reinterpretation and reimplementation of the analyses, as well as for the use of simplified model results directly given by the collaborations.

## B.1. Reinterpretation of analyses

In the case of reinterpretation for models different than those provided by the experimental collaborations, the information needed primarily includes expected and observed exclusion lines along with their $\pm 1\sigma$ uncertainty, expected and observed upper limits in case of simplified models, efficiency maps and kinematic distributions as reported in the analysis. If the kinematics of the new model to be tested in the reinterpretation is similar to that of the original model provided by the collaboration, it will be straight-forward to rescale the results provided to match the new model cross-section using this information.

## B.2. Reimplementation of analyses

One of the important developments in recent years is an active development of software codes [84,85,187–190] necessary for recasting analyses. The aim of these codes is to provide a public library of LHC analyses that have been reimplemented and validated, often by the collaborations themselves. Such libraries can then be used to analyze validity of a BSM scenario in a systematic and effective manner. The availability of public libraries further facilitates a unified framework and can lead to an organized and central structure to preserve LHC information long term. The reimplementation of an analysis consists of several stages. Typically, the analysis note is used as a basis for the implementation of the preselection and event selection cuts in the user analysis code within the recasting frameworks. Signal events are generated, and passed through a parameterized detector simulation using software such as Delphes or PGS [103,191]. The reconstructed objects are then analyzed using the code written in the previous step, and the results in terms of number of events are passed through a statistical analysis framework to compare with the backgrounds provided by the collaborations.

In order to be able to effectively use such codes, it is important to get a complete set of information from the collaborations.

For what concerns the generation of the models, it is desirable to have the following items as used by the collaborations:

* Monte Carlo generators: Monte Carlo generators along with the exact versions used to produce the event files should be listed.

* Production cross sections: The order of production cross sections (e.g. LO, NLO, NLL) as well as the codes which were used to compute them should be provided. Tables of reference cross sections for several values of particle masses are useful as well.

* Process Generation: Details of the generated process, detailing number of additional partons generated.

* LHE files: selected LHE files (detailing at least a few events if not the entire file) corresponding to the benchmarks listed in the analysis could also be made available in order to cross check process generation. Experimental collaborations may generate events on-the-fly without saving the intermediate LHE file; we advocate that the cross-check of process generation is straight-forward if this information is present, so we encourage the generation of a few selected benchmark points allowing for a LHE file to be saved. Special attention should be paid to list the parameters which change the production cross section or kinematics of the process e.g. mixing angles.

* Process cards: Process cards including PDF choices, details of matching algorithms and scales and details of process generation. If process cards are not available, the above items should be clearly identified.

* Model files: For models which are not already implemented in MADGRAPH5 AMC@NLO, the availability of the corresponding model files in the UFO format [77] is highly desired. This format details the exact notation used in the model and hence sets up a complete framework. In case MADGRAPH5 AMC@NLO is not used, enough information should be provided in order to clearly identify the underlying model used for interpretations and reproduce the generation.

The ATLAS/CMS Dark Matter Forum provides most of the information needed within its Git repository [59].

Efficiency maps and relevant kinematic distributions as reported in the analysis should be provided, in a digitized format with clearly specified units. If selection criteria cannot be easily simulated through parameterized detector simulation, the collaborations should provide the efficiency of such cuts. Overall reconstruction and identification efficiencies of physics objects are given as an input to the detector simulation software. It is thus very useful to get parameterized efficiencies for reconstructed objects (as a function of the rapidity $\eta$ and/or transverse momentum $p_T$), along with the working points at which they were evaluated (e.g. loose, tight selection). Object definitions should be clearly identifiable. Digitized kinematic distributions are often necessary for the validation of the analysis so that the results from the collaboration are obtained, and so are tables containing the events passing each of the cuts.

The availability of digitized data and backgrounds is one of the primary requirements for fast and efficient recasting. Platforms such as HepData [186] can be used as a centralized repository; alternatively, analysis public pages and tables can be used for dissemination of results. Both data and Standard Model backgrounds should be provided in the form of binned histogram that can be interpolated if needed.
<figure><img src="image_105.png" /><figcaption>Fig. C.81. Cross section of the $pp \to H\chi\bar{\chi}$ process as a function of $g_{hZ'Z'}$ for $m_{Z'} = 100$ GeV (left) and $m_{Z'} = 1$ TeV (right). The fit functions are shown in the text.</figcaption></figure>

A detailed description of the likelihood used in order to derive the limits from the comparison of data to signal plus background should be given. This can be inferred from the analysis documentation itself, however direct availability of the limit setting code as a workspace in RooStats or HistFitter [192] is highly desirable.

Finally, the collaborations can also provide an analysis code directly implemented in one of the public recasting codes detailed above. Such codes can be published via INSPIRE [193] in order to track versioning and citations.

### B.3. Simplified model interpretations

Dark Matter searches at the LHC will include simplified model interpretations in their search results. These interpretations are simple and can be used for a survey of viability of parameter space. Codes such as [194–196] can make use of the simplified model results given in the form of 95% Confidence Level (CLs) upper limit or efficiency maps in order to test Beyond the Standard Model parameter space. As mentioned above, it will thus be extremely useful if the results are given in a digitized form that is easily usable by the theory community.

The parameter space of these models should be clearly specified. For example, for a simplified model containing dark matter mass $m_\chi$, mediator mass $M_{\text{med}}$ and couplings $g_\chi, g_q$ it will be very useful to have upper limits on the product of couplings $\sqrt{g_\chi g_q}$ or cross section times branching ratio as a function of $m_\chi, M_{\text{med}}$. Limits on visible cross sections of the simplified models considered for interpretations should be made available.

The usage of simplified model results relies on interpolating between upper limit values. In order to facilitate the interpolation, regions where large variation of upper limits is observed should contain denser grid, if a uniform grid over the entire plane is not possible. For simplified model involving more than three parameters (two masses and product of couplings), slices of upper limits in the additional dimensions will be necessary for reinterpretation.

As already mentioned in the introduction to this Chapter, acceptance and efficiency maps for all the signal regions involved in the analysis should be made available. These results are not only useful for model testing using simplified models but also to validate implementation of the analysis. Information about the most sensitive signal regions as a function of new particle masses is also useful in order to determine the validity of approximate limit setting procedures commonly used by theorists.

## Appendix C. Additional details and studies within the forum

### Further information for baryonic Z' model

#### Cross-section scaling

The dependence of the cross section of the $pp \to H\chi\bar{\chi} + X$ process on $g_{hZ'Z'}$ is shown in Fig. C.81. The curves have been fit to second-order polynomials, where $y$ is the cross-section and $x$ is the coupling $g_{hZ'Z'}$.

For $m_{\text{med}} = 100$ GeV, the fit function is:

$$y = -0.12 - 3.4 \times 10^{-3}x + 2.7 \times 10^{-4}x^2. \qquad (C.1)$$

For $M_{\text{med}} = 1$ TeV, the fit function is:

$$y = 0.0012 - 2.4 \times 10^{-7}x + 1.5 \times 10^{-7}x^2. \qquad (C.2)$$

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