Dark-Shower Signatures in Hidden Sectors

• Dark-shower signatures are distinctive signals arising from a confining dark sector where dark quarks radiate and hadronize into a spectrum of dark mesons and baryons.
• They produce a rich mix of visible and invisible decay products, manifesting as semi‐visible jets, emerging jets, SUEP events, and multiple displaced vertices in collider experiments.
• Advanced modeling and substructure techniques, including machine learning classifiers, are employed to robustly differentiate these signatures from standard QCD backgrounds.

Dark-shower signatures refer to distinctive collider and fixed-target experimental signals arising from parton-shower dynamics in a strongly-coupled, confining hidden sector (dark sector), frequently motivated by Hidden Valley models or non-minimal dark-matter scenarios. These phenomena generalize standard QCD-like jet production to hidden-sector dynamics in which dark quarks and gluons, communicating with the Standard Model (SM) through one or more mediators (portals), shower into a spectrum of dark hadrons. This hadronization yields final states with a rich variety of visible and invisible signatures, depending on the dark-sector spectrum and its decay pathways back to SM particles. Dark-shower signatures encompass a range of collider objects including semi-visible jets, emerging jets, SUEP (“soft unclustered energy pattern”) events, collimated lepton jets, and multiple displaced vertices, each requiring dedicated theoretical modeling and experimental analysis frameworks.

1. Theoretical Foundations: Hidden Sectors, Portals, and Dark Parton Showers

The archetype for dark-shower phenomenology is the addition of a confining gauge group, such as SU(N_d), to the SM, with its own set of “dark quarks” $\chi_a$ (or $q_D$) transforming non-trivially under this new gauge symmetry (Cohen et al., 2017, Albouy et al., 2022, Deliyergiyev, 2015). Above the dark confinement scale $\Lambda_d$, the gauge coupling $\alpha_d$ is perturbative; below $\Lambda_d$, $\alpha_d$ becomes large, forcing the dark quarks to form a tower of dark mesons ($\pi_d,\,\rho_d$) and baryons. In realistic models, interactions between the dark sector and the SM are suppressed, characterized by so-called portals:

• Vector portals (e.g., kinetic mixing of a U(1)$_D$ “dark photon” $A'$ with hypercharge, or $Z'$ mediators):

$$\mathcal{L}_{\text{portal}} = -\frac{\epsilon}{2}\,F'_{\mu\nu}B^{\mu\nu} + g_q Z'_\mu \bar q \gamma^\mu q + g_\chi Z'_\mu \bar\chi \gamma^\mu \chi$$

• Scalar portals (mixing with the Higgs boson)
• Higher-dimensional effective operators (e.g., contact terms)
• Axial-vector or topological portals

When a hard process (e.g., at the LHC or a B-factory) produces a dark quark with $p_T \gg \Lambda_d$, it radiates dark gluons—analogous to a QCD parton shower—and fragments into multiple dark hadrons. Depending on the interplay of portal couplings and mass spectrum, these dark hadrons may be prompt, displaced, long-lived, or invisible on detector timescales (Cohen et al., 2017, Bernreuther et al., 2022, Bernreuther et al., 27 Oct 2025, Cheng et al., 16 Jan 2024).

2. Event Topologies and Experimental Signatures

The distinctive signature of a dark shower is determined by the mixture and lifetime of the dark hadron species:

1. Prompt Decays (QCD-like Final States): All dark hadrons decay promptly via the portal to SM quarks/leptons, creating jets closely resembling QCD jets both in energy flow and hadronic content (Buckley et al., 2022, Cohen et al., 2020).
2. Semi-Visible Jets: A fraction $r_{\rm inv}$ of dark hadrons are invisible (i.e., collider-stable, such as stable dark pions) while the rest decay promptly to SM particles. The resulting jets are hybrid objects containing both visible SM hadrons and invisible dark-matter particles, often yielding missing transverse energy ($E_T^{\rm miss}$) aligned with the jet axis rather than anti-aligned as in conventional mono-jet signatures. This topology gives rise to suppressed charged-particle multiplicity and broader jet-energy profiles compared to QCD jets (Cohen et al., 2017, Albouy et al., 2022, Bernreuther et al., 2020).
3. Emerging Jets: If some dark hadrons are long-lived on detector scales (displaced, but not completely invisible), they decay inside the detector to SM hadrons/electrons—producing jets with a spatially non-uniform charged-track distribution (emerging from secondary vertices) (Albouy et al., 2022).
4. Fully Invisible Showers: All dark hadrons are collider-stable, yielding only large $E_T^{\rm miss}$ recoiling against initial-state radiation (ISR).
5. Soft Unclustered Energy Patterns (SUEP): When the dark sector is almost conformal or the hadronization temperature is low, one may observe a sphere of low-$p_T$, high-multiplicity hadrons, characterized by isotropic energy flow (Albouy et al., 2022).
6. Multi-Vertex Topologies in Beam-Dump and Intensity-Frontier Experiments: Multiple displaced vertices within a single event—arising from the decay of several non-prompt dark vector mesons ($\rho_D^0$)—enable unambiguous discrimination against single LLP or minimal dark-photon scenarios (Bernreuther et al., 27 Oct 2025, Bernreuther et al., 2022).

3. Quantitative Parametrization and Theoretical Tools

Central to collider modeling is the “semi-visible jet” paradigm (Cohen et al., 2017), with a simplified-parameter space:

• $\Lambda$: Contact/intermediate scale
• $\Lambda_d$: Dark confinement scale (typically $\sim10$–$100$ GeV)
• $\alpha_d$: Dark gauge coupling at the hard scale
• $M_d$, $m_\chi$: Dark hadron and quark masses
• $r_{\rm inv}$: Invisible fraction of dark hadrons,

$$r_{\rm inv} = \frac{\langle \# \text{ stable hadrons } \rangle}{\langle \# \text{ total hadrons } \rangle}$$

These inputs—along with detailed decay widths, lifetimes, and mixing angles for portal particles—enable simulation using the HiddenValley module in Pythia 8 (Albouy et al., 2022, Buckley et al., 2022, Bernreuther et al., 27 Oct 2025). Benchmarking studies scan $r_{\rm inv}$ in $[0,1]$ to interpolate between QCD-like jets and mono-jet signatures.

The hadronization procedure is typically modeled by the Lund string fragmentation, with fragmentation and mass parameters either taken from QCD or scaled by $(\Lambda_d/\Lambda_{\mathrm{QCD}})^\text{power}$ (Buckley et al., 2022, Bernreuther et al., 2022).

4. Discriminating Observables and Jet-Substructure Methods

Analyses seeking to distinguish dark showers from QCD backgrounds and other SM processes require robust, IRC-safe jet substructure observables:

• Energy-Correlation Functions (ECFs):

$$e_n^{(\beta)} = \sum_{i_1<\ldots<i_n} \left(z_{i_1}\cdots z_{i_n}\right) \prod_{a<b} \theta_{i_a i_b}^{\beta}$$

• Energy-Flow Polynomials (EFPs):

$$\mathrm{EFP}_G = \sum_{i_1 \ldots i_N} z_{i_1}\cdots z_{i_N} \prod_{(k,\ell)\in \mathrm{Edges}(G)} \theta_{i_k i_\ell}$$

• N-subjettiness, Les Houches Angularity, Girth, $p_T^D$, Quark-Gluon Discriminants, etc.
• Machine Learning Classifiers:

Dedicated neural-network taggers, especially dynamic graph convolutional networks (DGCNNs) working on “particle cloud” input, have demonstrated superior performance in recognizing semi-visible jets relative to both jet images (CNNs) and Lorentz-layer networks. In (Bernreuther et al., 2020), a DGCNN achieved a background rejection $1/\epsilon_B=608\pm38$ at $\epsilon_S=0.3$, compared to $137\pm5$ (CNN) and $220\pm14$ (LoLa).

Statistical discrimination exploits combined multi-observable likelihood ratios (LLRs) and receiver-operating characteristic curves to optimize separation, e.g., combining EFPs with $C_2$ or $\tau_{21}$ can yield a factor $\sim2$–$5$ improvement at $\epsilon_S\sim50\%$ (Buckley et al., 2022).

Systematic uncertainties in modeling (especially hadronization and dark fragmentation) are significant; for the canonical two-point energy correlator $e_2^{(\beta)}$, the uncertainty envelope from parton-shower, hadronization, and parametric choices is explicitly quantified in (Cohen et al., 2020).

5. Experimental Search Strategies and Sensitivity Projections

Analyses targeting dark showers employ several key selection and search strategies:

• Inclusive jets + $E_T^{\rm miss}$:

Preselection with $E_T^{\rm miss}>200$ GeV, jets with $p_T>250$ GeV, and often a lepton veto. Partition events by $\min \Delta\phi(\text{jet}_i,\,E_T^{\rm miss})<0.4$ (semi-visible “signal” region) versus $>0.4$ (QCD background suppression) (Cohen et al., 2017).

• Dijet and Transverse-Mass Resonance Searches:

Dijet bump-hunts target low-$r_{\rm inv}$, while $M_T$ bump-hunts using “fat” jets probe intermediate $r_{\rm inv}$.

• Displaced-vertex analyses:

Particularly in beam-dump and $e^+e^-$ experiments (SHiP, Belle II), searches for multi-vertex events with invariant-mass and spatial/temporal correlation constraints exploit the multiplicity and boosted nature of dark vector mesons (Bernreuther et al., 27 Oct 2025, Bernreuther et al., 2022).

• Tagger-enhanced analyses:

Embedding DGCNN-based semi-visible jet taggers into event selection can improve cross-section sensitivity by more than an order of magnitude and probe mediator–quark couplings an order below that accessible by standard searches (Bernreuther et al., 2020).

• Emerging/SUEP analysis:

Emerging jets are selected via displaced tracks/multivertex tagging, while SUEP events are identified by high-multiplicity, nearly isotropic track patterns (Albouy et al., 2022).

• Background prediction:

Monte Carlo for $Z$/W+jets, $t\bar t$, QCD (MadGraph, Pythia, Delphes), and data-driven control regions (e.g., photon+jets with veto) are essential for robust systematic estimation (Cohen et al., 2017, Albouy et al., 2022).

The table below summarizes typical collider search topologies and region-of-interest variables:

Topology	Key Observables	Search/Selection
Semi-visible jets	$r_{\rm inv}$, $E_T^{\rm miss}$, $\Delta\phi$	Jet+$E_T^{\rm miss}$, substructure cuts
Emerging jets	Track multiplicity vs. radius, displaced tracks	Vertexing, prompt/displaced fractions
SUEP	High multiplicity, isotropy, low $p_T$	Multiplicity, ring isotropy, lack of ISR jet
Multi-vertex (BD)	Number of vertices, vertex mass/timing	Event-level $n$-vertex topology

In LHC projections, for $M_d=10$ GeV and $r_{\rm inv}\sim0.5$, the contact-operator scale $\Lambda$ can be excluded up to $\sim2.5$ TeV with $13$ TeV, $37$ fb$^{-1}$ data; s-channel $Z'$ searches extend up to $M_{Z'}\sim3$ TeV (for $g_q=0.1,g_\chi=1$) (Cohen et al., 2017). At Belle II and SHiP, displaced-vertex sensitivity extends to $\epsilon\sim10^{-6}$–$10^{-4.5}$ for $m_{\rho_D}\sim0.2$–$5$ GeV with coverage of multi-vertex events and dark matter parameter space (Bernreuther et al., 27 Oct 2025).

6. Systematic Uncertainties, Modeling Limitations, and Future Prospects

All collider and fixed-target analyses are strongly sensitive to uncertainties in dark-sector hadronization and fragmentation. Theoretical systematics arise from:

• Ab initio modeling in Pythia 8’s HiddenValley module, including parameters of the Lund string model, assumed degeneracy or mass spectrum of dark hadrons, and portal structure (Albouy et al., 2022, Buckley et al., 2022, Cohen et al., 2020).
• Absence, until recently, of alternate frameworks (cluster models as in Herwig) for cross-validation (Buckley et al., 2022).
• Unknowns in the mapping between simplified-model parameters and the true spectrum of stable/unstable states, especially where large portal coupling or additional mediator physics is involved (Deliyergiyev, 2015, Chen et al., 2018).

Benchmarks recommend varying hadronization parameters over wide ranges, quantifying one-sigma “envelope” uncertainties on all observables (Cohen et al., 2020). Experimental implementation should fold these systematics directly into signal efficiency and discovery significance calculations.

Future directions proposed include:

• Data-driven control regions for backgrounds in novel search regions (e.g., low $\Delta\phi$ in MET-based searches, photon-veto sidebands) (Cohen et al., 2017).
• Further refinement of jet substructure or particle-level machine-learning taggers, including DGCNN architectures (Bernreuther et al., 2020).
• Development of multi-object or multi-vertex triggers at intensity-frontier experiments (Bernreuther et al., 27 Oct 2025, Bernreuther et al., 2022).
• Exploiting jet-quark chirality sensitivity of shower profiles (e.g. distinguishing chiral from vector dark-matter models with the cumulative energy profile $\Psi(r)$) (Chen et al., 2018).
• Dedicated displaced-object detectors (e.g. MATHUSLA, FASER) for high-multiplicity, long-lived scenarios (Cheng et al., 16 Jan 2024).

7. Relation to Dark Matter and Cosmology

A considerable fraction of dark-shower scenarios is constrained or motivated by dark matter relic density and astrophysical observations:

• Stable dark pions ($\pi_D$) or baryons may constitute the cosmological dark matter, with direct detection suppressed by higher-dimensional form-factor effects (e.g., nuclear charge radius), often yielding direct-detection cross-sections well below the neutrino floor, $\sim10^{-52}$ cm$^2$ (Cohen et al., 2017).
• The $3\to2$ or $3\to\rho_D\to$SM annihilation mechanisms for dark pions can be probed across cosmologically interesting mass/coupling ranges, especially in the sub-GeV window at intensity-frontier experiments (Bernreuther et al., 27 Oct 2025).
• Complementarity of collider and non-collider probes is highlighted: the LHC, Belle II, LHCb, and SHiP together provide coverage of decay lifetimes from sub-millimeter scale (EWPT, LHCb) to multi-meter scales (Belle II, SHiP) (Bernreuther et al., 2022, Albouy et al., 2022).

In conclusion, dark-shower signatures constitute a rich and organized framework for exploring non-minimal dark sectors at the LHC, fixed-target, and intensity-frontier experiments. The interplay of parton-level production, showering/hadronization, and portal-induced decays—mapped quantitatively into observable collider objects and supported by machine-learning and substructure techniques—provides robust channels for discovering, disentangling, and characterizing hidden strong dynamics and its possible connection to dark matter.