Search for inelastic dark matter in events with two displaced muons and missing transverse momentum in proton-proton collisions at $\sqrt{s}$ = 13 TeV

A search for dark matter in events with a displaced nonresonant muon pair and missing transverse momentum is presented. The analysis is performed using an integrated luminosity of 138 fb$^{-1}$ of proton-proton (pp) collision data at a center-of-mass energy of 13 TeV produced by the LHC in 2016-2018. No significant excess over the predicted backgrounds is observed. Upper limits are set on the product of the inelastic dark matter production cross section $\sigma$(pp $\to$ A' $\to$ $\chi_1$ $\chi_2$) and the decay branching fraction $\mathcal{B}$($\chi_2$ $\to$ $\chi_1 \mu^+ \mu^-$), where A' is a dark photon and $\chi_1$ and $\chi_2$ are states in the dark sector with near mass degeneracy. This is the first dedicated collider search for inelastic dark matter.

In this Letter, we present a novel search for DM production in LHC proton-proton (pp) collisions that targets displaced-decay signatures.Specifically, we probe an inelastic DM (IDM) model [29][30][31] that postulates the existence of at least two states in the dark sector (the lightest of which is the stable DM) accompanied by a dark photon that kinetically mixes with the SM hypercharge [32].In inelastic coupling scenarios, these states cannot scatter elastically with other particles (e.g., nucleons) [31].These models predict a flavor-mixing off-diagonal vector coupling between the states and the dark photon, such that both states are produced simultaneously.A small mass gap between the lighter and heavier states leads to a compressed phase space and increased lifetime of the heavier state, thus producing LLP decay signatures in the CMS detector.Such models can account for the observed thermal-relic abundance [29], which is the density of dark matter energy left over from the early evolution of the universe, while evading the increasingly stringent experimental constraints set by other DM searches.Previous studies [29,[33][34][35] have placed bounds on the IDM production cross section for lighter dark matter masses (<1 GeV) by reinterpreting existing results from previous experiments [36][37][38][39] or via fixed-target experimental setups.The work described here is the first dedicated collider search for IDM, which provides new sensitivity to heavier DM masses (≈3-80 GeV) and to displaced nonresonant dimuon production.This is achieved via the use of a dedicated displaced muon reconstruction algorithm and optimized event selection criteria, including isolation requirements.The signal selection efficiencies may be as low as 10 −4 for low DM masses and highly displaced signal hypotheses, but the predicted IDM production cross sections can be as large as a few picobarns.The pioneering sensitivity achieved in this work to the unique finalstate topology comprising low-p T , displaced, and nonresonant muon pairs foreshadows the increasing attention devoted to more complex scenarios of new physics, given the absence of conclusive evidence for resonances or other singular phenomena incompatible with the SM.The techniques presented here will enable novel searches targeting models of DM with a rich structure and other compressed-spectra models such as supersymmetry models featuring coannihilation [40,41].Tabulated results are provided in the HEPData record for this analysis [42].
The process of interest includes a dark photon A ′ that is produced in the pp collision and recoils from an initial-state radiation jet, which is needed for efficient triggering on the event.The A ′ decays promptly to two states χ 1 and χ 2 with a near mass degeneracy, as illustrated in Fig. 1.The light DM state χ 1 (with mass m 1 ) is stable and not detected.The heavy state χ 2 (with mass m 2 ) travels a measurable distance before decaying to another χ 1 and a pair of SM particles via an off-shell dark photon.The mass splitting between the two states ∆ ≡ m 2 − m 1 is relatively small, between 10-40% of m 1 , resulting in decay products that are "soft" (transverse momentum p T ≲ 15 GeV) and have small angular separation.Here we focus on final states with muons because of the powerful reconstruction and identification tools available for displaced muons and the higher purity of the reconstruction compared to that of soft electrons.The escaping χ 1 particles are collimated with the soft muons and lead to sizable ⃗ p miss T , defined as the projection onto the plane perpendicular to the beam axis of the negative vector momentum sum of all reconstructed objects in an event.Its magnitude is referred to as p miss T .Previous CMS, ATLAS, and LHCb searches for processes with dimuon signatures [16, 17, 20-22, 24, 26, 43] are not as sensitive to this class of models, for various reasons.Some analyses rely on the detection of dimuon resonances [16,22,24,26], whereas muons in IDM are not resonantly produced.Others must compromise on either requiring higher muon p T [17,20,26,43] to keep the background and trigger rates manageable, or requiring muons to be prompt [21,24].However, such selections also remove a significant fraction of IDM signal-like events, which feature both soft and displaced muons.Finally, searches for monojet signatures could have some sensitivity to IDM models for muon displacements sufficiently high such that both muons are not reconstructed, but there would be a large background from multijet processes, as well as low signal production at the high muon-pair displacement.The signal sensitivity would therefore be small.The required signature including displaced muons allows backgrounds to be controlled without the need for a higher p miss T threshold, which would greatly reduce the efficiency to reconstruct the IDM signal.The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. The solenoid volume contains a silicon pixel and strip tracker (extending in radius from 4 to 110 cm and covering the range of pseudorapidity |η| < 2.4), a lead tungstate crystal electromagnetic calorimeter, and a brass-scintillator hadron calorimeter, each composed of a barrel and two endcap sections.Forward calorimeters extend the η coverage provided by the barrel and endcap detectors.Muons are detected in gasionization detectors embedded in the steel flux-return yoke outside the solenoid, extending radially from about 4 to 7 m and covering the range |η| < 2.4.A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [44].A particle-flow (PF) algorithm [45] aims to reconstruct and identify each individual particle in an event, with an optimized combination of information from the various elements of the CMS detector.
The analysis is carried out with data collected by the CMS Collaboration in 2016-2018 with a total integrated luminosity of 138 fb −1 .Simulated samples of signal and background events are used to optimize the event selection.Signal samples with dimuon decays are generated with MADGRAPH5 aMC@NLO v.2.6.0 [46,47] at leading order in quantum chromodynamics (QCD) and injected into PYTHIA v8.226 [48] for fragmentation and parton shower modeling.Motivated by Ref. [29] and other sources, we select parameters consisting of m 1 in the range 3-80 GeV, m A ′ = 3 m 1 , ∆ = {0.1,0.4}m 1 , cτ in the range 1-1000 mm, and α D = {0.1,0.4}.Here, cτ is the proper decay length of χ 2 and α D is the coupling strength of the U(1) D in the dark sector.These five parameters fix the kinetic mixing coefficient ϵ, which controls the amount of mixing between the dark photon and the SM hypercharge.The CUETP8M1 underlying event tune [49] is applied to 2016 samples and the CP5 tune [50] to 2017-2018 samples.Two parton distribution function sets are used: NNPDF3.0 [51] (2016) and NNPDF3.1 [52] (2017-2018).Additional pp interactions in the same or adjacent bunch crossings (pileup) are also simulated, with a frequency distribution matching that in data.Finally, the detector response is simulated with GEANT4 [53] and identical reconstruction algorithms are applied to collision and simulated samples.
The event selection exploits the unique features expected from IDM: a pair of collinear, soft, and displaced muons collimated with ⃗ p miss T .The muons are too soft to pass the trigger selection, so at least one energetic jet is required to boost the DM particles and enhance the p miss T spectrum.Candidate events are first selected with triggers with a minimum p miss T threshold of 120 GeV at the trigger level and 200 GeV in the offline selection, constructed excluding muons [54].The leading jet in the event is required to have p T > 80 GeV and |η| < 2.4.To accommodate additional initial-state emissions, only one other jet, with p T > 30 GeV and |η| < 5.0, is allowed per event.The jet p T requirement removes low-momentum jets produced in pileup interactions.Limiting the number of jets reduces the dominant background from events with jets produced through the strong interaction, referred to as QCD events, while retaining approximately 70% of the signal yield.To suppress top quark backgrounds, events are vetoed if any jets are identified as originating from a bottom quark, based on the loose working point of the DEEPCSV algorithm [55,56].The leading (subleading) jet must be azimuthally separated from ⃗ p miss T by at least 1.50 (0.75) rad.These selections further suppress backgrounds by ensuring that the DM system is well isolated in the event.
Muons are reconstructed with a specialized algorithm designed to remain efficient even for large displacements of the muon-pair vertex of up to several meters from the luminous region.This displaced standalone algorithm (DSA) uses only information from the muon system and does not require muons to originate from the interaction point [20,57].In Fig. 2, the DSA reconstruction efficiency for a representative signal sample is compared to that of the standard global reconstruction algorithm [57], which requires both tracker and muon detector information.The efficiency is calculated vs. the distance v xy in the transverse plane between the primary vertex (defined as the vertex with the largest value of summed charged particle p 2 T ) and the muonpair vertex.The efficiency remains high even when the muon-pair vertex lies beyond the inner radius of the muon detector planes.
The baseline muon selection requires at least two identified DSA muons.The identification criteria comprise >12 hits across ≥2 different muon detector planes (and >18 hits if no hits are found in the muon detector endcaps), track fit quality χ 2 /d.o.f < 2.5, p T resolution σ p T /p T < 1, p T > 5 GeV, and |η| < 2.4.The efficiency to identify such a DSA muon (about 90%) is measured with three different data samples providing complementary coverage of the kinematic phase space: cosmic ray muons (as a proxy for displaced muons), muon pairs from Z boson decays (high p T ), and muon pairs from J/ψ meson decays (low p T ).This efficiency is compared to that of the corresponding simulated sample and the yearly efficiency ratio is parametrized as a function of muon p T , η, and transverse impact parameter d xy (defined as the closest distance in the transverse plane between the track trajectory and the main vertex), and applied as a correction to simulated events [26].A cosmic ray muon veto is implemented by discarding events containing at least one pair of back-to-back DSA muons.At least one pair of oppositely charged DSA muons must form a well-reconstructed vertex (using a Kalman filtering algorithm [58]) as inferred from a vertex fit with quality χ 2 /d.o.f < 4 and a relative position resolution of about 2-3% (1%) at low (high) displacements.The muon pair with the smallest χ 2 value is chosen.The overall efficiency to correctly assign muon charges is greater than 97% because of the low-energy spectrum of IDM muons.Finally, we require the muons to be collimated with ∆R ≡ √ (∆ϕ) 2 + (∆η) 2 < 0.9, and the dimuon p T to be azimuthally aligned with the p miss T in the event, ∆ϕ miss µµ ≡ |∆ϕ(⃗ p miss T , ⃗ p µµ T )| < 0.5.These requirements remove more than 90% of the electroweak SM backgrounds without incurring a significant loss of signal efficiency.
The DSA muons have higher reconstruction efficiencies but degraded momentum resolution and impact parameter resolution compared to muons reconstructed with the standard algorithm.This is due to the smaller lever arm and lack of tracker information.To recover some of the performance at lower displacements, a match-and-replace procedure is implemented whereby DSA muon system tracks that are found close to muons reconstructed with the PF algorithm (PF muons) are replaced with the latter.The matching requirement is ∆R < 0.2 at the location of the innermost muon spectrometer track hit.The baseline requirement of two DSA muons in the event selection removes ambiguities in the PF-DSA matching.The vertex is then refitted and again required to pass the χ 2 /d.o.f < 4 threshold.We divide the signal region (SR) into three categories (0-2) based on the number of PF-DSA matches found.Highly displaced signal events fall in the zero-match category, since typically no PF muons are reconstructed, while signal events with slightly displaced muons fall in the one-or two-match categories, like most SM backgrounds.This categorization thus further enhances the search sensitivity for displaced signal models in the zero-match category.
Backgrounds passing the event selection consist mainly of three types: QCD events with genuine or misidentified muons and p miss T arising from jet mismeasurements (dominant in all match categories); multijet W events where the W boson decays to a muon and a muonneutrino and a misidentified muon is selected (contributing to the one-match category); and multijet Z events where the Z boson decays to two neutrinos and two misidentified muons are selected (contributing to the zero-and one-match categories).Misidentified muons may be associated with pileup interactions or the underlying event and therefore not correlate with the hard scatter in the event, or they may arise from instrumental deficiencies.No backgrounds from cosmic ray muons or other noncollisional contributions are expected because of the highly selective event requirements, the cosmic ray veto, and the use of p miss T triggers.The absence of such backgrounds was checked with a dedicated sample of events collected with noncollisional triggers, which require particles traversing the CMS detector while simultaneously vetoing any beam particles crossing the down-and upstream ends of the detector.
A modified matrix ("ABCD") method is employed to estimate the backgrounds, relying on two independent variables to discriminate between signal and background.The two-dimensional plane formed by these variables is divided into four bins (A-D).Because the variables are independent, all four bins can be described using only three parameters: a normalization rate plus a vertical and a horizontal transfer coefficient.The final degree of freedom is then used to fit the signal rate across all bins [59].
To maximize sensitivity, different variables are used in each match category.All three match categories use the minimum d xy of the two muons, referred to as min-d xy .In the one-and two-match categories, the second discriminating variable is the relative PF isolation I rel PF of the min-d xy muon, defined as the p T sum of all photons, charged and neutral hadrons found within a ∆R < 0.4 cone of a muon, divided by its p T .Unlike in QCD events, where muons originate from parton fragmentation and hadronization processes, muons in IDM should be isolated.In the zero-match category, where no PF muons are identified, the second ABCD variable is ∆ϕ miss µµ .The assumption of independence between all variables is extensively checked, as described in the next paragraph.The simultaneous fit accounts for the presence of signal in all ABCD bins, but a higher concentration is expected for high min-d xy , low I rel PF , and small ∆ϕ miss µµ .To validate the background estimation procedure and to optimize the ABCD binning, a multijet validation region (VR) enriched with backgrounds and devoid of signal is defined by inverting the requirement on the number of jets in the event, demanding at least three jets.Simulation studies show that QCD multijet events are the dominant background source in all match categories, comprising about 80% of the expected backgrounds both in the SR and in the VR.The shapes of all observables used in the ABCD procedure were found to agree within uncertainties between the VR and the SR, and between data and MC simulation.Several sets of ABCD bins are defined in the VR and the agreement between predicted and observed yields in bin D (the bin with the smallest backgrounds) is confirmed for each set.We choose a grid of bin boundaries centered around the optimal bins in each match category, for a total of 31 tests.The ensemble of control tests shows no statistically significant deviation between prediction and observation.Since these closure tests indicate that the observables chosen for the background validation are not correlated (within the statistical precision of the method), no systematic uncertainties are applied to the background prediction in the analysis, which is entirely dominated by statistical uncertainties.The optimal ABCD binning (reported in Table 1 for each match category) is determined by requiring at least three events in each of at least three bins of the ABCD plane, which is needed for a reasonable background estimation in the fourth bin.
Table 1: Definition of ABCD bins and yields in data, per match category.The predicted yield in the bin with the smallest backgrounds (bin D) is extracted from the simultaneous four-bin fit by assuming zero signal, which corresponds to (Obs.B × Obs.C)/(Obs.A) in this limit.Uncertainties in the predicted yields are purely statistical and determined from the propagation of Poisson uncertainties in the various bins.The dominant signal uncertainties in the analysis are either statistical, arising from a low selection efficiency for some signal hypotheses, or systematic due to imperfect knowledge of efficiencies, energy corrections, and the integrated luminosity [60][61][62].The total signal systematic uncertainties averaged over all years are approximately 20, 30, and 40% for the zero-, one-, and two-match categories, respectively (with a yearly breakdown shown in Table 2).These are applied uniformly to all signal hypotheses, unlike the statistical uncertainty, which depends on the signal efficiency of each hypothesis.The observed yields in data are used to perform a simultaneous fit to the four ABCD bins in each match category, and the fit results are found to be consistent with the background-only hypothesis, as illustrated in Table 1.Specifically, the measured (predicted) yields are 2 (1.2 ± 0.6), 0 (0.5 ± 0.2), and 0 (0.5 ± 0.2) in the zero-, one-, and two-match categories, respectively.Figure 3 shows the measured min-d xy distribution in the two-match category, for the subset of events passing the isolation requirement (B and D bins).To derive the background prediction shown in Fig. 3, the observed distribution in bins A and C is normalized to the sum of yields counted in bins B and D under the no-signal assumption.
We assess 95% confidence level (CL) limits on the product of the DM production cross section and decay branching fraction σ(pp → A ′ → χ 2 χ 1 ) B(χ 2 → χ 1 µ + µ − ) using a modified frequentist criterion CL s with the likelihood ratio test statistic [63,64].The upper limits are shown in Fig. 4 as a function of the mass m 1 of the DM state and the interaction strength y ≡ ϵ 2 α D (m 1 /m A ′ ) 4 .This choice of parametrization allows the relevant variables to be scaled with the thermal-relic abundance in a straightforward way, as explored in Ref. [29].The exclusion curves (in blue and magenta) depend more strongly on the choices of α D and of m A ′ than the cross section limits because of the resonant enhancement arising from the mixing between the A ′ and Z bosons when m A ′ ≈ m Z .Results are shown for m A ′ = 3 m 1 , which is chosen for compatibility with the target thermal-relic DM abundances as determined by cosmological observations.Regions of parameter space are excluded in both mass-splitting scenarios: for ∆ = 0.1 m 1 , values of y above ∼10 −7 -10 −6 and above ∼10 −8 -10 −7 , depending on the choice of α D , are excluded at m 1 = 3 and 80 GeV, respectively.For ∆ = 0.4 m 1 , the relative exclusion sensitivity is weaker because of the smaller production cross sections, which scale as 1/∆ 5 when all other parameters are fixed.Only the region near the A ′ -Z resonance is excluded in this scenario.
In summary, a search has been presented for inelastically coupled dark matter with a unique final-state signature including a soft, displaced muon pair collimated with the missing transverse momentum vector.The analysis is performed using proton-proton collision data produced by the LHC at a center-of-mass energy of 13 TeV and collected with the CMS experiment in 2016-2018.The data sample corresponds to an integrated luminosity of 138 fb −1 .Control samples in data are used to predict the background, and no significant excess is observed over ). Solid (dashed) curves denote the observed (expected) exclusion limits at 95% CL, with 68% CL uncertainty bands around the expectation.Regions above the curves are excluded, depending on the α D hypothesis: α D = α EM (dark blue) or 0.1 (light magenta).The sensitivity is higher in the region near m 1 ≈ 30 GeV or m A ′ ≈ 90 GeV because of the A ′ mixing with the Z boson in that mass range.standard model expectations.Upper limits are set on the product of the DM production cross section and decay branching fraction into muons as a function of DM mass m 1 and interaction strength.This is the first dedicated collider search for inelastic dark matter and it significantly expands the sensitivity to m 1 above the GeV scale.

Figure 1 :
Figure 1: Feynman diagram of IDM production and decay in pp collisions.The heavier state χ 2 can be long-lived, and decays to χ 1 and to a muon pair via an off-shell dark photon A ′ .

Figure 2 :
Figure 2: Simulated muon reconstruction efficiency of standard (blue squares) and displaced (red circles) reconstruction algorithms as a function of transverse vertex displacement v xy .The two dashed vertical gray lines denote the ends of the fiducial tracker and muon detector regions, respectively.

Figure 3 :
Figure 3: Measured min-d xy distribution in the two-match category, after requiring the min-d xy muon to pass the isolation requirement I rel PF < 0.25 (i.e., the B and D bins of the ABCD plane).Overlaid with a red histogram is the background predicted from the region of the ABCD plane failing the same requirement (the A and C bins), as well as three signal benchmark hypotheses (as defined in the legends), assuming α D is equal to the electromagnetic (EM) fine-structure constant α EM .The red hatched bands correspond to the background prediction uncertainty.The last bin includes the overflow.

Figure 4 :
Figure 4: Two-dimensional exclusion surfaces for ∆ = 0.1 m 1 (left) and 0.4 m 1 (right), as functions of the DM mass m 1 and the signal strength y, with m A ′ = 3 m 1 .Filled histograms denote observed limits on σ(pp→ A ′ → χ 2 χ 1 ) B(χ 2 → χ 1 µ + µ −).Solid (dashed) curves denote the observed (expected) exclusion limits at 95% CL, with 68% CL uncertainty bands around the expectation.Regions above the curves are excluded, depending on the α D hypothesis: α D = α EM (dark blue) or 0.1 (light magenta).The sensitivity is higher in the region near m 1 ≈ 30 GeV or m A ′ ≈ 90 GeV because of the A ′ mixing with the Z boson in that mass range.

Table 2 :
Systematic uncertainties in the analysis.The jet uncertainties are larger in 2017 because of noise issues with the ECAL endcap.The tracking inefficiency in 2016 is caused by the unexpected saturation of photodiode signals in the tracker.The first two rows give the uncertainty per PF muon.Thus, for the first row, the contributions are 0%, 5%, and 10% in the 0-, 1-, and 2-match categories, respectively.The third row lists the DSA displaced reconstruction (reco.)systematic uncertainty.