Monojet Signatures from Heavy Colored Particles: Future Collider Sensitivities and Theoretical Uncertainties
Abstract
In models with some colored particle that can decay into a dark matter candidate , the hadron collider process jets gives rise to events with significant transverse momentum imbalance. In the limit that the colored particles and dark matter candidates masses are very close, the signature becomes monojetlike, and experimental limits from searches at Large Hadron Collider (LHC) become much less constraining. In this paper, we study the current and anticipated experimental sensitivity to such particles at the HighLuminosity LHC at TeV with ab of data and the proposed HighEnergy LHC at TeV with ab of data. We estimate the reach for various scenarios of heavy colored particles , including spin0 and particles in the triplet and octet representations of QCD. The identification of the nature of is very important to identify the physics scenario behind the monojet signature. Therefore we also study the dependence of the cross section and of observables built from the leading jet in the process on the quantum numbers of . Using the stateoftheart Monte Carlo suites MadGraph5_aMC@NLO+Pythia8 and Sherpa, we find that when these observables are calculated at nexttoleading order (NLO) in QCD with parton shower matching and multijet merging, the residual theoretical uncertainties associated with the renormalization, factorization, and parton shower scales, as well as those associated with parton distribution functions, are comparable to differences observed when varying itself. We find, however, that the precision achievable with nexttonexttoleading order (NNLO) calculations, where available, could resolve this dilemma.
Keywords:
Standard Model, BSM physics, LHC, Jets, QCDKEKTH2050, IPPP/18/31
1 Introduction
The nature of dark matter (DM) remains one of the most outstanding mysteries in the particle physics today. Among the many possible particle candidates known Feng:2010gw (), weakly interacting massive particles (WIMP) are arguably the most theoretically motivated and wellstudied scenarios; for a recent review, see Abercrombie:2015wmb (). WIMPs with masses in the range of 10 GeV  20 TeV Plehn:2017fdg () can be stable on the age of the universe, and once they are in thermal equilibrium in the early universe remain so, even after decoupling occurs. Moreover, such stable particles are naturally present in many beyond the Standard Model (BSM) frameworks. For example: In weakscale supersymmetric (SUSY) theories Nilles:1983ge (); Haber:1984rc (), if one assumes Rparity conservation, then then lightest SUSY particle is stable, and hence is a potential component of DM. In the universal extra dimension (UED) models Appelquist:2000nn (), the lightest KaluzaKlein excitations of neutral electroweak bosons can be viable DM candidates.
Notably, a number of these new physics scenarios also involve additional heavy colored particles, , that couple to DM candidate(s). Hence, if and DM are kinematically accessible at the Large Hadron Collider (LHC), or its potential successors, such as the High Energy (HE)LHC, then it may be possible to study DM in a laboratory setting. In particular, once produced and if allowed, can decay into DM and SM particles leading to a plethora of interesting signatures at the LHC.
At hadron colliders, search strategies for these hypothetical colored particles usually involve investigating jets and leptons produced in association with finalstate DM candidates manifesting as large missing transverse energy (). In the context of simplified SUSY models, such signatures are now strongly constrained by LHC data if has a mass around 1 TeV and below Sirunyan:2017mrs (); Sirunyan:2017xse (); Aaboud:2017aeu (); Aaboud:2017bac (). Such constraints on , however, can be circumvented. One of the most celebrated examples of this is the compressed spectrum scenario Martin:2007gf (); Fan:2011yu (); Murayama:2012jh (). In this situation, the DM and new colored particles have a small mass splitting. Consequently, the visible decay products in the DM+SM process do not have sufficient momenta to be readily distinguished from SM backgrounds. In other words, the compression of the mass spectrum constrains the visible decay products of to possess such low momenta that they fail experimental selection criteria. This leads to significantly smaller selection and acceptance efficiencies, and hence significantly weaker bounds on heavy particle masses.
Even though a compressed mass spectrum represents a special corner of a typical BSM parameter space, the most attractive feature of this situation is that it allows relatively light, colored DM partners in light of presentday LHC data. This is particularly true for stops () and gluinos () in SUSY Schofbeck:2016oeg (); Sirunyan:2018iwl (); CMSPASSUS17005 (); ATLAS:2016kts (). If a compressed scenario is realized in nature, then one can experimentally resolve the soft, i.e., low , visible decays of by recoiling against a relatively hard, i.e., high , electroweak or QCD radiation that, in its own right, is sufficiently energetic to satisfy trigger criteria. One such process, shown diagrammatically in Fig. 1 and the focus of this study, is the inclusive monojet plus collider signature^{*}^{*}*Note that unlike the exclusive monojet signature, the current search strategy for the inclusive signature permits topologies with up to four analysisquality jets atlasconf2017060 ()..
Were evidence for the new particle established at the LHC, or a successor experiment, it would be crucial to determine ’s properties, especially its mass, spin, and color representation. Generically, such a program would involve investigating various collider observables that can discriminate against possible candidates for . For example: cross sections are highly sensitive to the aforementioned spacetime and internal quantum numbers. Consider the cases of a scalar top (color triplet), a spin1/2 top partner (color triplet), and spin1/2 gluino (color octet). For a fixed mass, i.e., , the pair production cross sections for these particles exhibit the hierarchy
(1.1) 
Conversely, for a fixed cross section, i.e., , one finds that
(1.2) 
This implies, however, that were a monojet cross section measured, the result could be replicated by different scenarios by a simple tuning of mass . In another way, one cannot constrain the mass of from cross section measurements alone without first asserting its color representation and spin. Quantitatively, this is more nuanced due to fact that leading order (LO) calculations are poor approximations for QCD processes, even when using sensible scale choices. Using Ref. Alwall:2014hca (), one can easily verify that, like the top quark Nason:1987xz (); Beenakker:1988bq (), QCD corrections at nexttoleading order (NLO) increase the production cross section by for TeVscale . This is the case at both and 27 TeV, and despite scale uncertainties at LO and NLO spanning and , respectively, for . Moreover, it is wellknown that nexttonexttoleading order (NNLO) corrections are nonnegligible for SM top production Czakon:2013goa (). It is also known that such large theoretical uncertainties can greatly limit the interpretation of the experimental results, particularly in searches for socalled topphilic dark matter Arina:2016cqj ().
The situation, however, is more hopeful following the advent of generalpurpose, precision Monte Carlo (MC) event generators. With software suites such as Herwig Bellm:2015jjp (), MadGraph5_aMC@NLO+Pythia8 (MG5_aMC@NLO+PY8) Alwall:2014hca (); Sjostrand:2014zea (), and Sherpa Gleisberg:2008ta (), automated event generation at NLO in QCD with parton shower (PS) matching is now possible for both SM and BSM Degrande:2014vpa () processes. Not only can one now readily include potentially important corrections to cross sections normalizations, but parton showers augment fixed order predictions with resummed corrections to at least the leading logarithmic (LL) level. As a consequence, such observables like the associated jet multiplicity in the monojet process, an exclusive observable that is critical to search strategies, is automatically modeled at LO+LL accuracy. This is the lowest order at which the quantity is qualitatively correct. In light of the availability of such sophisticated technology, one is now in position to systematically investigate the impact of QCD corrections on the inclusive monojet process.
In this report, we perform such a dedicated precision study on the inclusive monojet signature in the context of a compressed mass spectrum. As mentioned, observables associated with this process are highly sensitive to the mass, spin, and color representation of the mediating states. Hence, we consider benchmark models with representative mass, spin and color configurations for , with . Our study is aimed at the HLLHC, assuming at TeV, and the HELHC, assuming at TeV. We show how the monojet search strategy can be used to identify the nature of the heavy colored particle () from the dependence of the leading jet in pair production. We also estimate the precision required to distinguish these new physics scenarios. We quantitatively discuss various sources of theoretical uncertainty, including event generator dependence. Although we do not discuss the second jet distribution intensively in this paper, azimuthal angle correlation of the first and second jet contains information of the spin of , as discussed in Hagiwara:2013jp (); Mukhopadhyay:2014dsa ().
The remainder of this study continues in the following manner: In Sec. 2 we provide indept detail of our computational setup. In Sec. 3, we discuss observed and expected sensitivity of monojet searches at present and hypothetical future facilities, and address various theoretical uncertainties in Sec. 4. A brief outlook on the impact of this work is discussed in Sec. 5, and conclude in Sec. 6.
2 Computational and Theoretical Setup
Systematic studies of QCD radiation in the production of hypothetical, TeVscale colored particles are now possible due to the availability of precision, generalpurpose MC event generators. In practice, this nontrivial task is handled by using several individually published formalisms and software packages that have largely been integrated into a single framework or wellspecified tool chain sequence. In this section, we describe our computational and theoretical setups for modeling production with various associated jet multiplicities at LO+PS and NLO+PS in MG5_aMC@NLO+PY8 and Sherpa. For numerical results, readers can go directly to Sec. 3.
The section continues as follows: In Sec. 2.1, we briefly summarize the we consider from representative BSM scenarios. In Sec. 2.2, we enumerate the several methods for incorporating additional QCD radiation into production that we employ, and briefly note their main features and formal accuracies. We describe our setup for MG5_aMC@NLO+PY8 and Sherpa, respectively, in Secs. 2.3 and 2.4, and our detector simulation in Sec. 2.5. In Sec. 2.6, we summarize the SM inputs.
2.1 Framework for New Heavy Colored Particles
In this analysis, we consider three benchmark BSM candidates for : a stop squark , a gluino , and a fermionic top partner . To model these states in collisions at our desired accuracy, we use the NLO in QCDaccurate Universal FeynRules Object (UFO) Alloul:2013bka (); Degrande:2011ua () model libraries available from the FeynRules model database ufo_nlo_database (). The counterterms required for NLO computations and contained in these libraries are generated with FeynRules Alloul:2013bka (), using NLOCT Degrande:2014vpa () and FeynArts Hahn:2000kx (). For illustrative purpose, we choose three mass values for , namely = 400 GeV, 600 GeV, and 800 GeV. We note that as the spacetime and quantum numbers for are identical to those of the SM top quark, several publicly available calculations can be adapted in straightforward ways for . This includes total cross section predictions for inclusive production at NNLO in QCD, which we obtain using the HATHOR package Aliev:2010zk ().
We treat the decay in the narrowwidth approximation, thereby decoupling its production from its decay into an invisible particle . For each , the particle nature of the DM particle is chosen in accordance with its underlying theory: For , is a neutral scalar and are decayed to the twobody final state , where is a light QCD quark. For , is a neutral fermion and are decayed via an offshell top quark to the fourbody final state, . For , is a neutral fermion and are decayed to the threebody final state . We enforce a compressed mass spectrum by fixing a small mass gap between and to be . Since the mass gap is (relatively) small, the SM decay products of are forced to be (relatively) soft. Hence, the SM decay products of fail the criteria needed to be identified as the leading jet.
The choices of , their relevant quantum numbers and decay path, and the corresponding UFO library references are summarized in Table 1.
2.2 MultiLeg Matching and Merging Prescriptions
The collider signature considered in this work is characterized by the presence of a high jet recoiling against the system. It is thus necessary to include at least one QCD radiation at the matrix element level beyond what is modeled in Bornlevel production. With presently available MC technology, this can be achieved in different ways and at various formal accuracies. We now briefly describe the several prescription used in this study.

Leading Order Multijet Merging: The LO multijet/multileg merging techniques Lonnblad:2001iq (); Mangano:2006rw (); Lonnblad:2012ng () outline how parton shower emissions can be augmented with full matrix elements. The emissions are classified according to their hardness, i.e., , and in terms of a dimensionful variable . Emissions above a hardness threshold are described at LO accuracy using the appropriate matrix elements while preserving the allorder resummation accuracy of the parton shower below . In this work we use the MLM scheme Mangano:2006rw () as implemented in MadGraph5_aMC@NLO 2.6.0 (MG5_aMC@NLO) Alwall:2014hca (). We take into account matrix element corrections to pair production in association with up to two QCD partons. While genuine (and higher) corrections are included via this procedure, the calculation remains formally LO accurate (LO+LL after parton showering) due to missing virtual corrections. Some of the earlier studies on monojet spectra with LO multijet merging technique using then available Madgraph/MadEvent v4 Alwall:2007st (), we refer Alwall:2008va (); Alwall:2008qv (); LeCompte:2011fh (); Dreiner:2012gx (); Dreiner:2012sh ().

Production at NexttoLeading Order with Parton Shower Matching: The accuracy of fixed order (FO) matrix element calculations at NLO can be combined with resummed parton showers at LL by means of NLO+PS matching techniques Frixione:2002ik (); Nason:2004rx (); Frixione:2007vw (), such as the MC@NLO formalism Frixione:2002ik (). In this approach, the LO matrix element for an extra hard, wideangle QCD emission in the final state is naturally included as part of the FO correction. Extra soft and/or collinear emissions enter through the parton shower. Potential double counting of soft/collinear contributions is avoided by the use of additional counter terms. It is worth noting that leadingjet observables in this calculation are at most LO+LL accurate, but are nonetheless welldefined at all .

Production at NexttoLeading Order with Parton Shower Matching: In order to achieve NLO+LL accuracy for leadingjet observables, the above NLO+PS matching technique must be applied to the process. This requires explicitly introducing a regularizing selection on the leading jet at the matrix element level. Since no Sudakov form factor is present for this jet, the selection must also be well above the Sudakov shoulder of the inclusive system to ensure that the FO is perturbatively valid.

NexttoLeading Order Multijet Merging: The LO multijet merging technique described above can be extended to describe jet observables at NLO precision for jets above a hardness scale . Analogous to the LO case, matrix element merging with parton shower matching at nexttoleading order, known colloquially as MEPS@NLO, is achievable by introducing additional allorders/resumed Sudakov form factors for each NLOaccurate matrix element in consideration. We use an extension of the CataniKraussKuhnWebber (CKKW) Catani:2001cc (); Krauss:2002up (); Schalicke:2005nv () merging formalism as implemented in Sherpa by Refs. Gehrmann:2012yg (); Hoeche:2012yf (). In the following, we employ MEPS@NLO multijet merging with up to one or two jets, meaning that samples will contain up two or three real radiations, respectively, beyond the lowest order process before parton showering. As in the LO(+LL) case, while corrections are present in this calculation, the final result remains formally NLO(+LL) accurate.
2.3 Event Generation in MadGraph5_aMC@NLO + Pythia8
Cross section calculations and event generation with accuracy up to NLO in QCD are handled using MG5_aMC@NLO. For signal processes, we use the NLOaccurate UFO libraries described in Sec. 2.1 and listed in Table. 1 as inputs to MG5_aMC@NLO. Within the MG5_aMC@NLO framework, oneloop virtual corrections are evaluated numerically via MadLoop Hirschi:2011pa () and matched with real emissions using the FrixioneKunsztSigner (FKS) subtraction formalism Frixione:1995ms (), as implement by Ref. Frederix:2009yq (). Decays of are then handled at LO accuracy with MadSpin Artoisenet:2012st (). The central value for the renormalization scale and factorization scale is set to
(2.3) 
where is the scalar sum of the transverse energy of the final state partons and .
In Sec. 3, where we discuss experimental searches and sensitivity to in monojet searches, we impose an analysislevel selection on the leading jet in an event. More specifically, we set the ptj variable in MG5_aMC@NLO’s run_card.dat file to ptj (and jet radius ). Then, to enhance yields at relatively high , we generate events by binning the phase space in . After preparing event samples for a particular , we apply higher until the statistical uncertainty of the MC samples is no longer negligible. At this point, we prepare another event sample based on and apply the procedure iteratively. For the samples with the highest , we apply exponential biasing on at the eventgeneration level to enhance the tail of jet distributions.
After the pair have been decayed, events are passed to Pythia 8.2.26 Sjostrand:2014zea () for parton showering and hadronization. We choose a shower starting scale small enough such that light, colored finalstate partons in the matrix element remain the hardest emissions in the full process, if the parton exists. Namely, the parton shower is restricted to operate below the scale Alwall:2014hca ()
(2.4) 
Here, is the minimum distance measure as calculated with the algorithm Catani:1993hr (); Ellis:1993tq () (for ) over all momentum recombinations of light, colored partons during the clustering phase in fixed order event generation. The events are essentially categorized by whether or not a hard emission is present.
To quantify and estimate the size of missing, higher order QCD corrections, we compute the threepoint scalevariation envelope. This is obtained in the usual fashion, i.e., by varying discretely and jointly the factorization and renormalization scales and over the range,
(2.5) 
Where necessary, we also consider the uncertainty associated with the parton shower starting scale. We quantify this by discretely and independently computing the scale variation over the range,
(2.6) 
2.4 Event Generation in Sherpa
For crossvalidation of the fermionic top quark partner , we employ Sherpa 2.2.4 Gleisberg:2008ta (). At LO in QCD, arbitrary BSM models can be simulated through Sherpa’s generic UFO model Degrande:2011ua () interface Hoche:2014kca (). At NLO, processes involving can be simulated by slightly modifying the default SM model file, and setting the top quark mass to the mass of . For the decay into a scalar dark matter particle, an additional decay vertex is added using the methodology of Ref. Hoche:2014kca (). Treelevel matrix elements of the calculation are provided by Sherpa’s inhouse matrix element generators AMEGIC Krauss:2001iv () and COMIX Gleisberg:2008fv (). Oneloop amplitudes are treated by interfacing with OpenLoops Cascioli:2011va (). Parton showermatching is performed according to the MC@NLO formalism Frixione:2002ik (); Gleisberg:2007md (); Hoche:2012wh (), using Sherpa’s CataniSeymour subtractionbased shower procedure Catani:1996vz (); Schumann:2007mg ().
To study potential improvements to modeling fermionic top quark partners, we also employ multijet merging at NLO in QCD with Sherpa. To account for additional high QCD emissions at the matrixelement level, beyond what is already present in inclusive production at NLO, we include the NLOaccurate matrix elements for the processes
(2.7) 
We merge these samples with the fully inclusive sample following the MEPS@NLO prescription Gehrmann:2012yg (); Hoeche:2012yf (). In this scheme, the nominal values for the factorization, renormalization, and parton shower scale are determined through a backward clustering procedure that maps higher multiplicity configurations to a configuration. We set all scales to the invariant mass of the top partner pair. As a nominal value for the merging scale , we set . For parton showering we employ one of Sherpa’s dipole showers, which is published in Schumann:2007mg ().
In addition to MEPS@NLO merged samples we also simulate at NLO+PS with Sherpa. We do not add matrix element corrections to these samples beyond what is already present at NLO+PS. For the generation of the process, we use the scale schemes,
(2.8) 
where and denote the system’s invariant mass and transverse momentum.
2.5 Detector simulation and object reconstruction
For fast detector simulation, we use Delphes 3.3.3 deFavereau:2013fsa () with the default ATLAS card. Jets are constructed from calorimeter tower elements using Fastjet 3.2.1 Cacciari:2011ma (), according to the anti jet clustering algorithm Cacciari:2008gp () with jet radius . Analysisquality jets are required to satisfy the fiducial and kinematic criteria,
(2.9) 
Events are then accepted or rejected based the monojet selection criteria discussed in Sec. 3.
2.6 Standard Model Inputs
We assume active/massless quark flavors and a diagonal CabbiboKobayashiMaskawa (CKM) quark mixing matrix with unit entries. Relevant SM inputs used in our study include,
(2.10) 
The evolution of parton distribution functions (PDFs) and the strong coupling constant are extracted using the LHAPDF 6.1.6 Buckley:2014ana () libraries. As discussed in Sec. 2.2, LO (NLO) multijet merging with two (one) additional partons accounts for new kinematic channels and configurations that first arise at NNLO. (In principle, all one would need to achieve NNLO accuracy are the missing the twoloop virtual corrections.) However, such contributions are already accounted for in the normalizations of NLO PDFs. Hence, to minimize potential double counting of initialstate contributions, we use the NNPDF 3.0 NNLO PDF set (lhaid=261000) Ball:2014uwa () for all signal process calculations. For the LO SM calculation in Sec. 3, we use the NNPDF 3.0 LO PDF.
3 Monojet Searches at the HL and HELHC
At hadron colliders, the term “monojets” represents a broad class of sensitive collider signatures and search strategies that assume varying degrees of particle multiplicity and inclusiveness. In this section, we consider specifically the inclusive monojet signature, as implemented by ATLAS during Run II of the LHC’s operations after collecting 36.1 fb of integrated luminosity at TeV atlasconf2017060 (). After discussing various sources of experimental and theoretical uncertainties, we report the observed and expected sensitivity of the channel at current and proposed colliders using the modified frequentist approach Read:2002hq (). The modelindependent limits derived in Sec. 3.1 are then applied in Sec. 3.2 to the heavy colored particles described in Sec. 2.1. The ability to determine and distinguish principle properties of is then discussed in Sec. 3.3.
3.1 Monojet Searches, LHC Data, and ModelIndependent Limits
IM  [GeV]  Expected SM Events with  Statistical Error  Total Error 

Total (Stat.+Sys.+Th.) Error  
1  496 (0.20%)  2.3 %  
2  371 (0.27%)  2.5 %  
3  270 (0.37%)  2.6 %  
4  200 (0.50%)  2.5 %  
5  113 (0.89%)  2.6 %  
6  68 (1.46%)  3.4 %  
7  45 (2.23%)  4.4 %  
8  30 (3.32%)  6.1 %  
9  22 (4.64%)  7.3 %  
10  15 (6.48%)  9.7 % 
The ATLAS and CMS collaborations have both reported on their search of early Run II data for anomalous events with significant transverse momentum imbalance and at least one energetic jet atlasconf2017060 (); sirunyan:2017jix (). For the case under consideration atlasconf2017060 (), the ATLAS collaboration has investigated two overlapping signal regions, categorized as exclusive modes (EM) and inclusive modes (IM), based on various thresholds spanning GeV to 1 TeV. The EM signal regions are defined in terms of binning. For example: signal region EM1 (EM5) selects for events with satisfying 250 GeV 300 GeV (500 GeV 600 GeV). The IM signal regions are defined in terms of minimum selections. For example: signal region IM1 (IM5) selects for events with satisfying 250 (500) GeV. Additionally, events are required to satisfy the following selection criteria:

At least one analysislevel jet with GeV and .

A maximum of four analysislevel jets satisfying GeV and .

An azimuthal separation of for each analysislevel jet and the vector.
For the remainder of this study, we focus on the IM monojet signal regions.
In Table 2, we display the expected number of SM (background) events passing all selection criteria with uncertainties (statistical and total) in each of the inclusive mode signal regions (IM1IM10), as reported by ATLAS atlasconf2017060 (). Nonstatistical errors include both experimental and theoretical uncertainties. Sources of systematic uncertainty estimated in Ref. atlasconf2017060 () include: dependencies on parton shower and PDF modeling, which span 0.7% to 0.8%; uncertainties in jet energy and scales, which range 0.5% (IM1) to 5.3% (IM10); jet quality and pileup descriptions additionally provide uncertainties ranging 0.8% to 1.8%; for more detailed discussions, see Ref. atlasconf2017060 (). We note that the total errors for IM15 are nearly flat, with 2.2% to 2.6%, indicating that these signal regions’ uncertainties are systematics dominated. Statistical and systematic uncertainties are much larger in the higher regions. However, with the HL phase of LHC, one expects to collect significantly more data that will correspondingly reduce statistical errors for the high regions. Additionally, the analysis’ control sample will also increase during the HL run, therefore also reducing systematic uncertainties. Thus, one anticipates that total uncertainties will shrink for future inclusive monojet searches at LHC.
IM  Observed limit [fb]  Expected limit [fb]  Scaled limit [] for ab  

2.5% Syst. Error  1% Syst. Error  
1  531  324  333  133 
2  330  194  187  75 
3  188  111  99  39 
4  93  58  54  22 
5  43  21  17  6.9 
6  19  9.8  6.4  2.6 
7  7.7  5.7  2.8  1.1 
8  4.9  3.4  1.2  0.5 
9  2.2  2.1  0.6  0.3 
10  1.6  1.5  0.3  0.2 
The nonobservation of data deviating significantly from SM predictions enables one to set modelindependent upper limits on the production cross section of new particles. In the second and third columns of Table 3, we tabulate the expected and observed limits at the 95% confidence level (CL) on the visible cross section^{†}^{†}†Explicitly, the visible cross section is defined as the product of total cross section, acceptance, and efficiency., respectively, for each IM signal region, as reported in atlasconf2017060 () with 36.1 fb of data. To address the prospect of the HLLHC with ab at = 13 TeV, we calculate the expected upper limits by scaling the number of events at the 13 TeV run of LHC. We choose two values of the total systematic uncertainty, namely 2.5% and 1%. The former is a pessimistic assertion that systematic uncertainties will not be reduced beyond present levels (see Table 2), even after 1520 years of LHC operations. The latter is an optimistic, but benchmark, assumption. The likelihoods of background only and signalplusbackground hypotheses are set as Gaussian, with a standard deviation set to the total uncertainty. We have checked that our likelihoods are in agreement with those reported by Ref. atlasconf2017060 () for fb. We report the scaled limits in the last two columns of Table 3. We now discuss the impact of these limits on the production of pairs in collisions.
3.2 HL and HELHC Sensitivity to Heavy Colored Particles
We now compare and apply the modelindependent upper limits on the cross sections derived in the previous section to the production of pairs at the LHC. In Fig. 2, we show the modelindependent 95% CL upper limits along with NLO+PSaccurate cross section for produced in association with a hard jet at the matrixelement level, viz. at NLO. (For details of our computational setup, see Sec. 2.) We overlay FO scale uncertainty, computed according to Eq. (2.5). For all cases of , we find that the scale uncertainty at NLO is a dominant source of uncertainty and hence take it as a representative measure of the total uncertainty. A dedicated and indepth discussion of this uncertainty is given in Sec. 4.
From inclusive monojet searches at TeV with of data, we find that the lower limits on masses stand at around 400 GeV for the fermionic top partner and 600 GeV for the gluino, while no constraint on stop masses is found within the range under consideration. We observe that in the high bins both the systematic and statistical experimental uncertainties play a crucial role. As argued in the previous section, one expects sensitivity to improve at the HLLHC due to a much larger dataset, leading to better control on both uncertainties. From the scaled limits, we find that fermionic top partners with masses 800 GeV, gluinos with 1000 GeV, and stops with masses 600 GeV, in a compressed spectrum scenario, can be excluded at 13 TeV with ab, using the inclusive monojet signature.
Along with more data, a possibility that can greatly push the sensitivity to heavy colored particles is increasing the beam energy of the LHC itself. Presently, community discussions are underway on upgrading the LHC’s magnet system to handle a centerofmass energy up to TeV conf27tev (). In light of this prospect, we briefly investigate the impact on the production of and the SM background, i.e., the dominant background of the monojet signature, and estimate the experimental reach of such a collider.
For representative masses, , we show in Fig. 3 the cross sections for the and processes as a function of the leading jet . More specifically, the cross sections are calculated as a function of a generatorlevel threshold on the light jet. We scale the cross section (doubledotdashed) by factor 1/20 so that the ratio of the curve to the signal cross section is normalized with respect to the postevent selection limit in Fig. 2. In the lower panel, we show the 27 TeVto14 TeV cross section ratios.
We briefly note that while the signal processes are obtained at NLO in QCD accuracy, the background is considered only at LO^{‡}^{‡}‡Stateoftheart highprecision calculations for process are already available in literature, see Lindert:2017olm ().. This is due to the fact that jet observables in events can possess very large QCD corrections due to the opening of new kinematic configurations Rubin:2010xp (). In the present case, the process with TeVscale indeed exhibits such behavior. The increase corresponds to the real emission of low bosons in high dijet events. However, these configurations have relatively small , and thus not relevant to our monojet study. We have checked that the QCD factor lies within 1.2 for events after requiring large missing momentum at the NLO+PS level. As a result, LO predictions for the process represent a more appropriate description of the background than at NLO, particularly after the aforementioned normalization procedure. In Table 4, we list the cross sections for the process at and 27 TeV for representative selections.
Returning to the lower panel of Fig. 3, one sees that the production cross section of increases faster than the SM background with increasing centerofmass energy. The enhancement follows from the welldocumented Martin:2009iq (); ArkaniHamed:2015vfh (); Mangano:2016jyj () growth in PDF luminosities for fixed partonic mass scales but increasing collider beam energy. Quantitatively, for TeV, the cross section increases by a factor of 10 with respect to the change of , while the fermionic top partner cross section for GeV increases by approximately , the gluino rate for GeV by , and the stop rate for GeV by . Although other sources of SM backgrounds for the monojet signature have been presently neglected, the signal over background ratio still increases significantly at higher collider energies due to the larger luminosity enhancement for the signal process than dominant SM backgrounds. Subsequently, the HELHC enables ones to investigate parameter regions that are not accessible at the LHC. We do emphasize, however, that ratios can change drastically if additional information is provided to enhance the separation of the signal events from the backgrounds. For example: proposals exist on how to utilize soft leptons, jets, and displaced vertices associated with decays of that can further reduce SM background rates Chakraborty:2016qim (); Nagata:2017gci (); Chakraborty:2017kjq ().
(GeV)  cross section (pb)  

13 TeV  14 TeV  27 TeV  
300  9.40  11.1  41.9 
400  2.59  3.11  13.4 
500  0.889  1.09  5.22 
600  0.350  0.438  2.36 
800  7.18  9.33  0.637 
1000  1.85  2.49  0.214 
1200  5.46  7.63  8.44 
1400  1.77  2.59  3.69 
1600  6.03  9.29  1.71 
We end this discussion by providing an estimate of the anticipated sensitivity of new colored particles at the 27 TeV HELHC. Here, we assume that the SM background is dominated by the process and simply scale the modelindependent 95% CL upper limit at TeV by the 27to13 TeV production cross section ratio. In other words, the SM background cross section at a given and , denoted as , is obtained from the relation
(3.11) 
In the above, is the LO cross section with at a collider energy . We further assume that the detector acceptance and efficiencies are the same at 13 and 27 TeV. This assumption is not as strong as one may anticipate in more general circumstances. The HELHC project proposes to refit, replace, and/or upgrade the current LHC magnet system and detector experiments. As the detector experiment caverns themselves cannot physically grow, one is forced to adopt a detector fiducial volume at 27 TeV that is largely unchanged from 13 TeV. Similarly, we also assume systematic uncertainties of 2.5% and 1%, the same considered in Sec. 3.1.
Arguably, the background modeling of a LO process appears naive at first glance. However, the dominant SM backgrounds for inclusive monojet searches are indeed electroweak boson production atlasconf2017060 (). Such processes possess an initialstate parton composition and color structure comparable to , and also exhibit a similar dependence on collider energy. Hence, as shown in Fig. 4, once the scaled limits for the SM backgrounds are determined, we can compare the predicted cross sections for NLO process and estimate the expected reach at the 27 TeV HELHC. We find that with ab, one is sensitive compressed spectra scenarios featuring fermionic top partners with masses GeV, gluinos with masses GeV, and stops with masses GeV.
3.3 Properties Determination of Heavy Colored Particles
We now turn to the possibility of extracting properties of the heavy colored particle from jet behavior within the monojet signature. As briefly discussed in the introduction, asserting color representation and spin of is required to infer information on its mass from cross section measurements (or limits). Consequently, a single cross section measurement of a particular monojet signal region does not help much in determining the nature of . For example: in Fig. 3, one sees that the production cross section for the process with GeV, for a top squark with 400 GeV, a gluino with 900 GeV, and a fermionic top partner with 600 GeV are roughly within 5  10 fb of one another. However, despite this ambiguity, it is still possible to extract information from the cross section as a function of the leading jet , which can be measured directly, since it obeys a distinguishing pattern for each hypothesis. That the nature of is, in part, encoded in this observable reflects a nontrivial interplay between ’s mass, , its color representation and spin, and the dimensionless ratio . This interplay is what we now discuss.
The first discerning observation is that the cross sections do not depend on in a universal manner. Keeping to Fig. 3, one sees that while are the same at TeV for the configurations under consideration, the relative size of changes with . In other words, while follows an anticipated powerlaw of , with , the precise value of the exponent is dependent on the color and spin structure of . In a particular extreme, the gluino rate is the smallest (largest) of the configurations for smaller (larger) than , suggesting a smaller than for other . Information on can be extracted from by considering its ratio with respect to a benchmark . For example: for the benchmark process with , the 14 TeV cross section ratios at = 400 and 800 GeV are = 0.79 and 1.33, respectively, and similarly = 0.85 and 1.23. From this one can determine that the change in the cross section ratios over the range 400 GeV 800 GeV, is for gluinos and for fermionic top partners. Hence, cross section ratios for two different hypotheses is crucially dependent on the choice of .
In addition, one can also think about the physics case where there are multiple heavy particles making up the total cross section consistent with a lighter particle. Take for example that the ratio of is 0.21 at . Hence, five copies of with mass can mimic the cross section of a single with at At the however, The one and fivecopy scenarios then predict and , respectively, thus providing a means to check this potential degeneracy. Likewise, four copies of with can mimic the the cross section of at At however, one finds that . For the one and fourcopy scenarios, this leads to the predictions of and , respectively.
The different dependence on observed for gluinos, fermionic top partners, and stops arise from the fact that heavier particles give rise to harder, i.e. less steeply falling, distributions. The benchmark cases having significantly different input masses, with , and , and , and makes a significant impact on the dependence. To isolate this behavior, in Fig. 5 we plot the cross section double ratio as a function of ,
(3.12) 
In the top (bottom) single ratio, both cross sections are with respect to the mass but different . This has the effect of canceling overall color and kinematic factors while isolating logarithmic terms of the form . The double ratio, then, is a measure of this logarithmic dependence with respect to a baseline mass and minimum transverse momentum . For , we choose and , and plot for and . Quantitatively, one sees that the double ratio increases (decreases) by about 50% at GeV for GeV. This feature is universal for particles in the same color representation and follows from the nature of massless gauge boson emission in scattering processes.
To better understand this behavior, consider the process. The channel propagators gives rise, after phase space integration, to the aforementioned logarithms . In the context of parton shower resummation, this dependence can be interpreted as the likelihood of emitting an additional QCD parton with transverse momentum , i.e., the differential probability is proportional to . Hence, a fixed probability implies a fixed ratio, and indicates that increasing results in increasing commensurately. Qualitatively, the emission of higher QCD partons becomes easier for heavier because high emissions become relatively soft as increases. This results in a rightward shift of the socalled Sudakov shoulder Collins:1984kg (); Catani:1997xc (). For TeVscale particles, the rightward shift of what constitutes “soft” is known to be large; see for example Refs. Ruiz:2015zca (); Degrande:2016aje ().
The discrimination power of cross section ratios extends if one considers the additional dependency on a collider’s beam energy. In particular, we find that the cross section ratios shift from unity for GeV at TeV, to and 1.35 at 27 TeV. Hence, a measurement of the ratio of the cross section at the and 27 TeV with 30% accuracy can resolve our benchmark gluino, fermionic top parter, and stop scenario. Moreover, increasing the beam energy can also significantly improve the signaloverbackground ratio, thereby enabling measurements of the cross section for different minimum over a wide range of . Measuring the signal cross section for several at with 10% accuracy allows one to distinguish the benchmark scenario by narrowing down the mass of without information. In light of this, it is necessary to emphasize that theoretical predictions on total and differential cross sections, as well as their ratios, must have the requisite accuracy to make these measurements.
Lastly, we note that the slope of with respect to does not significantly depend on either the color structure or spin of . To show this, we display in Fig. 6 a second cross section double ratio,
(3.13) 
The structure of here is analogous to in Eq. (3.12), but differs in that the normalizing process is fixed to with GeV for TeV, and we vary in the upper ratio. Subsequently, while overall color and kinematic multiplicative factors cancel, the relative dependence of , , and scattering within an individual process does not cancel and is inherently dependent on the color representation.
From Fig. 6, one can observe that fermionic top partners and stops with same mass have the almost same slope over a wide range of . On the other hand, because of color and matrix element effects from gluinos, the ratio tend to decrease with but again independent to the mass of the heavy particle. For validation, we consider the case for scalar gluons, known as sgluons Kribs:2007ac (); Plehn:2008ae (); GoncalvesNetto:2012nt (), and overlay sgluon behavior on the same figure^{§}^{§}§The realization of sgluons with a compressed mass spectrum is beyond the scope of this paper.. As the plot suggests, the distribution of the hard radiation associated with the colored particle production follows universally for a given color representation of .
4 Theoretical Uncertainties of the Monojet Process
As investigated in Sec. 3.3, were one to discovery new colored particles at the LHC, or a potential successor experiment, a measurement of the cross section as a function of the leading jet’s minimum transverse momentum can help establish the quantum numbers of . Ascertaining such information, however, requires accurate BSM signal predictions. For the benchmark scenarios listed in Table 1, we find one needs theoretical uncertainties no larger than and , respectively, on the total inclusive cross section normalizations and on the change of the cross section for as a function of per . In this section, we discuss and quantify theoretical uncertainties associated with the monojet signal process. We particularly investigate (potential) sources of uncertainties when employing the stateoftheart MC suites MG5_aMC@NLO+PY8 and Sherpa.
To investigate theoretical uncertainties associated with the process, we focus on fermionic top partners . We choose this benchmark because of wide implementability across different event generators as well as its comparability to production in the SM. We compare several simulation setups and techniques at different levels of precision within QCD; see Sec. 2 for details. As the relevant signal topology in this work is characterized by the presence of hard QCD radiation recoiling against the system, our primary benchmark observable is the cross section as a function of the transverse momentum of the process’ leading, i.e., highest , jet .
We begin with Fig. 6(a), where we show the cross section as a function of as derived from the fully inclusive calculation at NLO+PS (dash), as obtained from the MC@NLO formalism. We also show the calculation at LO+PS (dot), as obtained with up to two additional matrix elementlevel jets via multijet merging. The curves are shown with their factorization and renormalization variation envelopes and are normalized to the respective nominal prediction. As discussed in Sec. 2, the leading jet distribution for both calculations is only LO accurate for comparable to the scale of the hard process. Hence, the scale uncertainties for are large, asymmetric, equal for the two calculations, and span approximately to . For smaller , namely , one observes that the two uncertainty envelopes begin to differ. Whereas the uncertainty for the multijet calculation reduce only slightly for decreasing , the MC@NLO uncertainty reduces to roughly the level. The difference originates from virtual corrections present in the NLO+PS calculation, which soften dependencies on , but are obviously absent in the LO+PS calculation. The Sudakovlike factor in the multijet merging prescription only partially reduces a dependence on by matching low QCD emissions in the hard matrix element with those in the PDF. Hence, even for observables that are formally of the same precision, the presence of all terms in the NLO+PS calculation leads to a smaller scale dependence at low than in the merged LO+PS calculation.
The small scale uncertainty observed for the lowest suggests high theoretical precision is achievable with NLO+PS computations obtained via the MC@NLO prescription. However, unlike pure FO calculations, calculations matched to parton showers possess the additional dependence on the parton shower starting scale . Within the MC@NLO formalism, controls whether the leading emission beyond the Born process is included in the FO matrix element or the allorders parton shower. Loosely speaking, QCD radiations with above (below) originate from the hard matrix element (parton shower). As pointed out in Refs. Hoeche:2011fd (); Jones:2017giv (), lowest orderaccurate observables, e.g., the distribution when the process is evaluated at NLO+PS, and processes that possess large virtual corrections suffer from ambiguities when choosing . This manifests as a strong dependence on , and hence a large uncertainty.
We assess this uncertainty in Fig. 8 by plotting the cross section, derived from the inclusive calculation at NLO+PS, as a function of . We assume multiplicative variations of the default parton shower scale, given in Eq. (2.6), but fix the factorization and renormalization scales to their central values. Rates are normalized to the fixed order NLO (fNLO) prediction. The variation of the NLO+PS result with respect to the default choice spans roughly , with the dependence increasing (decreasing) for larger (smaller) values of over the range of considered. The shower scale variation amounts to absolute deviations from the fNLO result up to about and . For vanishing , one should take caution in interpreting the vanishing shower scale uncertainty. In this limit, the FO calculation is unphysical. The FO calculation possess an integrable singularity at that leads to arbitrarily large cross sections. As a result, the ratio of the NLO+PSlevel cross section, a finite and physical quantity, to the fNLO cross section, the unphysical quantity, vanishes as .
In light of the large theoretical uncertainties in the cross section stemming from and , it is clear that the precision achieved with the aforementioned methods is insufficient for distinguishing candidates. To explore if such precision is yet still possible with presently available generalpurpose Monte Carlo technology, we consider the cross section obtained from the process itself at NLO+PS. In Fig. 6(b) we plot the normalized cross section with its envelope again as a function of