Hidden Valleys: Latent Sectors and Landscapes

• Hidden Valleys are latent sectors in physics and mathematics characterized by suppressed, hard-to-detect dynamics and intricate internal structures.
• In particle physics, they manifest as weakly-coupled hidden gauge sectors with distinctive collider signals such as displaced vertices and dark showers.
• In optimization and stochastic systems, hidden valleys denote rugged loss landscapes with spurious minima and deep intervals of low observable values.

A hidden valley is a concept that arises in several disparate areas of physics and mathematics, unified by the idea of structures, features, or regions that are either latent, challenging to probe directly, or have suppressed or subtle observable consequences. The meaning and significance of "hidden valleys" is highly context-dependent: in high-energy theoretical and experimental physics, particularly collider phenomenology and dark sector model-building, the term refers to sectors or gauge structures that are sequestered from the Standard Model and communicate via portals or weak couplings. In mathematical physics, most notably in the analysis of optimization and stochastic partial differential equations (PDEs), "hidden valleys" are connected components of sublevel sets—a landscape feature of the relevant functional or random field—often associated with local minima or with regions of anomalously low function values. This article provides a technical, multidisciplinary synthesis of hidden valleys as they arise in particle physics, collider phenomenology, machine learning optimization, statistical physics, and condensed matter systems.

1. Hidden Valleys in Particle Physics and Collider Phenomenology

The hidden valley (HV) paradigm in particle physics originated as a framework for extending the Standard Model (SM) with additional, "hidden" gauge sectors that can feature rich, low-mass dynamics but remain challenging to detect due to their weak or suppressed couplings to SM fields (0712.2041, Zurek, 2010, Pierce et al., 2017, Liu et al., 5 May 2025). The prototypical construction takes the form:

• Hidden gauge group $G_v$ (commonly non-Abelian, such as $SU(N_v)$) with new matter ("$v$‐quarks").
• Portal interactions enabled by heavy mediators (vector bosons $Z'$, Higgs portal, kinetic mixing).
• Confinement at a scale $\Lambda_v\lesssim 1$–$100$ GeV, leading to bound states ("$v$‐hadrons", e.g., dark pions, $\rho_v$, $\eta_v$, or glueballs) (0712.2041, Zurek, 2010).

Production and decay at colliders proceeds by:

1. Production of $v$‐quarks or glueballs via the portal (e.g., $pp\to Z'\to v\bar{v}$).
2. Hidden showering and hadronization into $N_v$ $v$‐hadrons, often resulting in high multiplicities (Liu et al., 5 May 2025).
3. $v$‐hadron decays back to SM particles, typically through the same suppressed portal (e.g., $\rho_v\to\mu^+\mu^-$), leading to observable signals such as narrow dilepton resonances, displaced vertices, or soft, high-multiplicity jets ("dark showers") (Born et al., 2023).

Experimental searches are challenged both by the suppression of portal-induced cross-sections (scaling as $g_{\text{portal}}^4/M^4$) and by the kinematics of $v$‐hadron decay:

• Signatures include high-multiplicity events, low thrust, spherical topologies, displaced lepton or hadron jets, and soft non-pointing decay products (0712.2041, Born et al., 2023).
• Dedicated strategies exploit vertex-based triggers and low-threshold tracking (e.g., LHCb’s two-DV or MET + "displaced shower" searches in CMS) to probe lifetimes $c\tau$ from $\sim$mm to $\sim$100 m (Pierce et al., 2017, Liu et al., 5 May 2025).

The HV formalism is also realized in extra-dimensional scenarios (Bunk et al., 2010), string theory–motivated quiver gauge theories (Cvetič et al., 2012), and minimal supersymmetric extensions with hidden $U(1)_x$ sectors (Chan et al., 2011), in which portal couplings and anomaly inflow play central roles in the phenomenology and mass protection of messenger fields.

2. Mathematical Structure: Hidden Valleys in Optimization Landscapes

In the topological analysis of non-convex optimization, particularly loss landscapes of neural networks, "hidden valleys" denote connected components of sublevel sets of the loss function that do not contain a global minimizer ("spurious valleys") (Venturi et al., 2018, Lin et al., 2020). Formally, for a parameterized loss $L(\theta)$, the $\alpha$-sublevel set is $\Omega_L(\alpha) = \{\theta: L(\theta)\leq\alpha\}$; a spurious valley is a path-connected component of $\Omega_L(\alpha)$ none of whose points achieve the global infimum.

A key result is that:

• For two-layer (one-hidden-layer) neural networks with activation $\sigma$, the absence or presence of spurious valleys is characterized by the intrinsic dimension of the function class $\text{span}\{\sigma(\langle w, x\rangle)\}$ (Venturi et al., 2018).
• For activations with finite upper intrinsic dimension (e.g., polynomials), networks with hidden width $p\geq \dim^*(\sigma, n)$ are free of spurious valleys.
• For commonly used non-polynomial activations (ReLU, sigmoid), the lower intrinsic dimension is infinite—spurious valleys can and do exist for adversarial data distributions, even as $p\to\infty$.
• However, high-risk spurious valleys are typically confined away from random initializations in wide networks, so gradient descent is often able to escape or avoid them in practice (Venturi et al., 2018).

For sparse neural networks, the location and structure of hidden valleys depends sensitively on the architecture:

• One-hidden-layer networks with sparse inputs but a dense final layer ("SD" topology) can be free of spurious valleys under mild overparameterization and structural conditions (see Theorems 1–4 in (Lin et al., 2020)).
• If the final layer is also sparse ("SS" topology), strict spurious valleys can arise, blocking gradient descent even when the network is otherwise overparameterized—empirically verified across numerous activations (Lin et al., 2020).

These results inform pruning and architecture design, emphasizing the importance of maintaining sufficient expressivity and connectivity in the final layer to avoid the proliferation of detrimental loss landscape valleys.

3. Hidden Valleys in Stochastic and Random Media

In statistical physics, the term "hidden valleys" describes spatial regions in random fields or solutions to stochastic partial differential equations (SPDEs) where observables are uniformly suppressed between rare, high-amplitude intermittent peaks. In the 1D stochastic heat equation with space–time white noise,

$$\partial_t u(t,x) = \frac{1}{2} \partial_x^2 u(t,x) + \sigma(u(t,x))\,\dot{W}(t,x),$$

"intermittency islands" are exponentially narrow, high peaks, while "hidden valleys" are vast intervals where $u(t,x)$ is exponentially small (Khoshnevisan et al., 2022).

Rigorous results:

• For initial data $u_0(x)\equiv1$, there exists with probability one a valley of length at least $\exp(A_1 t^{1/3})$ around the origin at time $t$, on which $u(t,x)\leq \exp(-A_2 t^{1/3})$ [(Khoshnevisan et al., 2022), Theorem 1.1].
• For initial data with subgaussian tails, the supremum of the solution decays super-exponentially on the whole real line, so the entire line becomes a valley as $t\to\infty$ [(Khoshnevisan et al., 2022), Theorem 1.2].
• The geometric distribution of valleys versus peaks is thus dual: valleys are macroscopic, exponentially long, and deep, while peaks are sparse and extremely local.

These findings refine the classical intermittency picture, quantifying both the extremal events (peaks) and the complementary extreme voids (valleys), indicating fractal or clustered geometric organization as $t\to\infty$.

4. Hidden Valleys in Condensed Matter: Spin and Valley Degrees of Freedom

In certain quantum and condensed matter systems, "hidden valleys" refers not to spatial sectors but to internal quantum degrees of freedom that are symmetry-protected or spatially sequestered. In multilayer transition metal dichalcogenides (e.g., 2H-WSe$_2$), the combination of strong spin–orbit coupling and crystal symmetry results in local, layer-resolved spin polarizations ("hidden spin textures") even when the global system retains inversion symmetry (Fanciulli et al., 2023).

Key features:

• Circularly polarized light can selectively excite bright excitons in well-defined momentum (K, K′) and spin valleys, generating nearly fully spin-polarized exciton populations in the topmost layer, observable via layer-selective time- and angle-resolved photoemission (Fanciulli et al., 2023).
• Intervalley scattering induces rapid ($\sim 60$ fs) decay of this bright-exciton spin polarization, while concurrent rapid formation of momentum-forbidden dark excitons provides a stable local spin reservoir whose polarization persists for over 1 ps.
• These "hidden valleys" of spin are operationally relevant for ultrafast spin injection and valleytronics in van der Waals heterostructures.

The physical origin is the alternating spin texture enforced by the stacking and symmetry of the 2H polytype, breaking spin degeneracy only locally within each layer but restoring it globally.

5. Portals, Phenomenology, and Implications

Hidden valley sectors can be technologically and phenomenologically moribund unless appropriate portal couplings are present. Portals can be renormalizable (dimension-4)—Higgs portal ($|H|^2\Phi$), kinetic mixing ($F_{\mu\nu}F'^{\mu\nu}$), heavy $Z'$ exchange—or higher dimension, generated at loop or tree level by integrating out heavy connector states (Zurek, 2010, Cvetič et al., 2012, Bunk et al., 2010).

The choice and structure of portals determine:

• Production cross sections and decay topologies at colliders; e.g., Drell–Yan–like $pp\to Z'^{(*)}\to v\bar{v}$ (0712.2041).
• Mediation of supersymmetry breaking or dark-matter interactions (through gauge, $Z'$, instanton, or axion portals) (Cvetič et al., 2012, Bunk et al., 2010).
• Astrophysical and cosmological signals, e.g., in black hole superradiance or axion–photon conversion (Dubovsky et al., 2010).

Even minimal kinetic mixing, or symmetry-protected but non-trivial anomaly–inflow structures (as in stringy hidden valleys), can have dramatic consequences for the viability, cosmological legacy, and observability of HV sectors (Cvetič et al., 2012, Bunk et al., 2010).

6. Landscape Phenomena, Model-Dependence, and Experimental Frontiers

A comprehensive hidden valley phenomenology must address:

• The interplay between mass gap, particle multiplicity, decay widths, detector geometry, and background rejection in experimental searches (Liu et al., 5 May 2025, Born et al., 2023).
• The importance of reporting and interpreting limits in the multidimensional parameter space $(\Lambda, r \equiv m_\pi/\Lambda, N_C, N_F)$, as the mapping from theory to signal is highly model-dependent (Liu et al., 5 May 2025).
• The risk of spurious valleys (in machine learning and mathematical physics contexts) if the model design, data distribution, or architectural constraints introduce hidden obstructions that are not immediately apparent from low-level observables or naive overparameterization arguments (Lin et al., 2020, Venturi et al., 2018).

Current and future colliders, as well as ultra-sensitive astrophysical and condensed matter probes, are expanding the experimental reach into hidden valley parameter space, necessitating the continuous refinement of both theoretical frameworks and experimental analysis strategies (Pierce et al., 2017, Liu et al., 5 May 2025).

7. Summary Table: Hidden Valleys Across Domains

Domain	Hidden Valley Manifestation	Key Mechanisms/Criteria
Particle Physics	Sequestered, confining gauge sector, low-mass bound states	Portals (Z', Higgs, kinetic), hadronization, displaced decays
Optimization/Loss Landscapes	Sublevel set components not reaching global minimum	Intrinsic dimension, architecture, overparameterization
Stochastic PDEs	Spatial intervals of vanishing field amidst intermittent peaks	Large deviation dissipation, exponential scaling of valley length
Condensed Matter (2D TMDs)	Layer-resolved hidden spin/valley polarization; subsymmetry-protected textures	Local crystal symmetry, spin–orbit coupling, selective photoexcitation

This domain-general overview underscores that "hidden valleys" are not merely a thematic label but denote a precise structural concept: sectors, components, or regions that are physically or functionally relevant but suppressed, protected, or otherwise occluded from direct detection, often resulting in non-trivial phenomenological, computational, or probabilistic implications.