Displaced Muon-Jets Technique

• Displaced Muon-Jets technique is a method to identify and reconstruct non-prompt, tightly collimated muon clusters from long-lived particle decays.
• It employs specialized tracking, vertexing, and muon clustering algorithms to detect spatially isolated muon signatures with large impact parameters.
• The approach enhances sensitivity to beyond Standard Model scenarios by establishing exclusion limits on lifetimes and cross sections using advanced data-driven techniques.

The displaced muon-jets technique refers to the experimental and analytical methodology developed to identify, reconstruct, and interpret the presence of highly collimated (small ΔR, typically < 0.1) clusters of muons that are spatially displaced from the primary proton–proton interaction vertex. Such topologies originate from the decays of long-lived neutral particles, often motivated by hidden-sector, Higgs-portal, or dark photon models, and have been the focus of dedicated searches at high-energy collider experiments such as ATLAS and CMS. The key features of this technique are the focus on non-prompt, spatially isolated muon clusters, use of advanced vertexing and track reconstruction adapted for large impact parameters, and statistical interpretation of limits as functions of lifetime and production cross section. The approach provides a critical window into new physics scenarios that evade standard prompt decay searches by exploiting unique displaced signatures.

1. Experimental Motivation and Signature Definition

The signature targeted by the displaced muon-jets technique is a set of two or more muons, highly collimated due to the decay of a boosted, light, long-lived particle (LLP), where the decay vertex is spatially separated from the primary interaction region. This separation can range from fractions of a millimeter up to several meters, depending on lifetime and boost. The theoretical motivation arises in extensions of the Standard Model—such as hidden valleys, dark photon scenarios, and Higgs-portal models—where non-Standard decays of the Higgs or other BSM states can yield long-lived intermediates decaying into muons. Characteristically, the opening angle between the muons can be estimated as: $$\Delta R_{\mu\mu} \simeq \frac{2 m_{\text{LLP}}}{p_{T,\text{LLP}}}$$ with typical values ΔR ~ 0.01 for light, GeV-scale LLPs with tens of GeV pT (Chang et al., 2016).

This experimental signature is distinct from QCD or prompt lepton-jets due to the spatial displacement of the decay vertex and the collimation of the muon pair or multiplet.

2. Reconstruction Techniques: Tracking, Vertexing, and Clustering

Reconstruction of displaced muon-jets leverages both inner detector (tracking) and outer detector (muon spectrometer) systems. Standard tracking algorithms are optimized for prompt tracks with small impact parameters ($d_0$); the displaced muon-jets technique therefore makes use of specialized algorithms that relax impact parameter requirements and reprocess unused hits (Collaboration, 2019).

Key components:

• Large Radius Tracking (LRT): Customized algorithms that include tracks with large transverse ($|d_0|$) and longitudinal ($|z_0|$) displacements from the beamline, with thresholds typically in the order of millimeters to centimeters (Collaboration, 2019).
• Vertex Fitting: Displaced tracks are clustered into candidate vertices using a χ² minimization, with relaxed quality and topology requirements to allow for non-pointing, spatially separated vertices (Collaboration, 2019, Chang et al., 2016).
• Muon-Jet Clustering: Muon candidates are clustered within small cones (ΔR < 0.1–0.3) according to their angular separation to form muon-jets (Collaboration, 2012, Chang et al., 2016).
• Pixel Hit Recovery: In case of merged muon signals (nearly overlapping tracks), pixel hit recovery algorithms extrapolate muon trajectories and search for shared pixel hits, compensating for low resolution or cluster merging in high-density regions (Dildick et al., 2015).
• Muon Spectrometer Use: For decays occurring beyond the calorimetry but within the MS, algorithms reconstruct “tracklets” from arranged MDT hits to form vertices without relying on prompt inner detector tracks (Collaboration, 2018, Collaboration, 2022).

3. Event Selection, Triggers, and Background Suppression

Displaced muon-jet analyses impose stringent event-level selections to suppress Standard Model backgrounds (e.g., prompt muon pairs from Drell–Yan, heavy flavor, and cosmic ray muons):

• Displacement Requirements: Candidate muons (or muon-jets) are required to be inconsistent with the primary vertex using impact parameter or vertex separation (e.g., transverse displacement $L_{xy}$, significance $|d_0|/\sigma_{d_0}$) (Collaboration, 2012, Dildick et al., 2015).
• Trigger Strategies: Inclusive muon triggers are made more flexible to accommodate displaced topologies. High-mass and low-mass dimuon triggers, reduced pointing constraints, and the use of MET or jet-based triggers in the outer detector enable collection of non-prompt muon pairs (Gershtein et al., 2019, Wulz, 2019).
• Isolation and Cosmic Ray Rejection: To address backgrounds from cosmic rays and beam-induced noise, cuts on timing, spatial vertex location, and isolation from other tracks/jets are essential (Collaboration, 2018).
• Statistical Subtraction/Background Modeling: Novel data-driven techniques, such as the P/M (plus-minus) approach using the 3D angle $\phi_{3\rm D}$ between the flight direction and reconstructed momentum, exploit symmetry in background distributions to cancel systematics, especially relevant in the mm–cm range of displacements (Santocchia, 27 May 2024).

4. Statistical Interpretation, Efficiency, and Lifetime Mapping

The efficiency for detecting displaced muon-jets is strongly dependent on decay location, lifetime, boost, and detector geometry. Signal expectations are computed as: $$N_{\rm signal} = \sigma(pp \rightarrow H) \times BR(H \rightarrow \text{hidden}) \times \mathcal{L} \times \epsilon(c\tau)$$ where $\mathcal{L}$ is the integrated luminosity and $\epsilon(c\tau)$ is the position-dependent efficiency (Collaboration, 2012). Varying $c\tau$ and comparing observed to expected event counts yields exclusion regions.

Statistical inference uses the CL${}_s$ technique to combine uncertainties and set 95% CL limits on $\sigma \times BR$ as a function of the mean lifetime. In some scenarios, the difference between event counts in regions enriched or depleted in signal (as per kinematic variables like $\cos\,\phi_{3\rm D}$) is used as the discovery discriminator (Santocchia, 27 May 2024).

5. Applicability: Physics Models and Collider Searches

The displaced muon-jet technique is sensitive to a broad class of models:

• Higgs Portal Models: Nonstandard Higgs decays, e.g., $h \rightarrow a a \rightarrow 4\mu$ (where $a$ is a light scalar or pseudoscalar) (Dildick et al., 2015, Chang et al., 2016). Final-state topologies include two 2μ-jets, one 2μ and one 4μ-jet, or two 4μ-jets.
• Dark Photon and Hidden Sector Models: Scenarios with long-lived neutral bosons decaying to pairs of muons (Collaboration, 2012, Dildick et al., 2015, Wulz, 2019).
• GMSB and Other BSM Models: Long-lived neutralinos decaying through $Z \rightarrow \mu^+\mu^-$ (Wulz, 2019).
• Model-Independent Limits: Cross-section times branching ratio limits on generic BSM neutral LLPs decaying to muon-jets.

Parameter space coverage extends from sub-millimeter up to meter-scale lifetimes, with varying efficiencies across the range. For example, assuming a 100% branching fraction, mean lifetimes of 1 mm – 670 mm can be excluded for $m_H$ = 100 GeV (Collaboration, 2012), and model-independent bounds on $BR$ as low as $\mathcal{O}(0.1\%)$ have been set using Run 2 datasets (Collaboration, 2022).

6. Methodological Developments and Analysis Innovations

Continued methodological innovations refine the displaced muon-jets technique:

• Advanced Triggers: Phase II CMS upgrades with track stub capability enable Level-1 triggering on displaced dimuons at low pT thresholds and large $d_0$, substantially increasing sensitivity to low-mass exotics (Gershtein et al., 2019).
• Neural Network and Machine Learning Classification: Deep learning tools are incorporated to distinguish prompt from displaced signatures, for instance via image-based calorimeter analysis and domain adaptation (Bhattacherjee et al., 2019, Collaboration, 2019).
• Displacement in the Sub-mm Region: Adaptive vertex fit and data-driven separation techniques allow sensitivity to lifetimes $c\tau \gtrsim 100\,\mu$m, with the capability to measure lifetimes using the separation between reconstructed DVs (Ito et al., 2017, Santocchia, 27 May 2024).

The availability of high-luminosity and acceptance, improved vertexing, and generalized selection approaches enables both the extension of exclusion limits and the ability to measure properties (such as lifetime) of putative signals.

7. Impact and Outlook

The displaced muon-jets technique has established robust constraints on models predicting long-lived neutral particles decaying to collimated muons by exploiting signatures that would evade traditional prompt searches. Enhanced reconstruction algorithms, advanced vertexing, improved triggers, and analytical tools have made it possible to probe mean lifetimes over several orders of magnitude, setting absolute constraints on branching ratios and cross sections well below the percent level in some scenarios (Collaboration, 2012, Collaboration, 2022).

A plausible implication is that as algorithms for displaced reconstruction, background estimation, and large-radius tracking continue to improve, the technique is expected to play an increasingly central role, enabling both model-specific and model-independent searches at the LHC and future colliders. In particular, data-driven methodologies such as the P/M approach in the sub-cm region (Santocchia, 27 May 2024) allow coverage of parameter space that had previously been inaccessible due to systematics, further enhancing the discovery potential for new physics beyond the Standard Model.