Sterile KK States in Warped Extra Dimensions

• Sterile Kaluza-Klein states are higher-dimensional excitations emerging from compactification, characterized by suppressed couplings to Standard Model fields.
• Their mass spectra exhibit double-indexed clusters due to nested boundary conditions and Bessel function solutions in multiply warped geometries.
• These modes can alter collider signatures and decay rates, offering new avenues for testing extra-dimensional models in particle physics and cosmology.

Sterile Kaluza-Klein (KK) states are higher-dimensional field-theoretic excitations that, due to their localization in the extra-dimensional sector and weak or vanishing coupling to the Standard Model brane, remain largely undetectable in ordinary 4D interactions. Their existence, spectra, and phenomenological roles vary significantly depending on the details of the compactification geometry, the types of fields involved, symmetries, and boundary conditions. Investigation of sterile KK states is crucial for understanding the implications of extra-dimensional models for collider searches, cosmology, and unified theories.

1. Definition and General Features

Sterile KK states arise as excitations in the tower of four-dimensional (4D) fields generated by higher-dimensional fields upon compactification, particularly those that do not carry Standard Model gauge charges or whose couplings are highly suppressed due to localization or boundary conditions. In multiply warped or generically compactified spacetimes, sterile KK states generally refer to modes associated with internal excitations of bulk fields that do not mix or interact strongly with brane-localized matter. In phenomenology, "sterile" is used to characterize such states as invisible or inert in standard detection channels yet potentially influential via gravitational or other suppressed couplings (Koley et al., 2010, Grard et al., 2010, Hinterbichler et al., 2013).

2. Kaluza-Klein Reduction in Multiply Warped Extra Dimensions

In six-dimensional, doubly warped geometries, the massive scalar field $\Phi(x, y, z)$ can be decomposed as: $$\Phi(x, y, z) = \sum_{n,p} \phi_{np}(x) \, \xi_n(y) \, \chi_p(z)$$ where $\xi_n(y)$ and $\chi_p(z)$ satisfy Sturm-Liouville-type eigenvalue equations with warp-factor-induced differential operators (Koley et al., 2010). Specifically, for the $y$-direction: $$\frac{1}{R_y^2} \frac{d}{dy} \left(a(y)^4 \frac{d\xi_n}{dy}\right) - m_p^2 a(y)^2 \xi_n = - m_{np}^2 \xi_n$$ and for the $z$-direction: $$\frac{1}{r_z^2} \frac{d}{dz}\left(b(z)^5 \frac{d\chi_p}{dz}\right) - m_\phi^2 b(z)^2 \chi_p = - m_p^2 \chi_p$$ Here, $a(y)=e^{-c|y|}$ and $b(z)=\cosh{k(\pi-z)}/\cosh(k\pi)$ are the warp factors along $y$ and $z$ respectively, with typically $c\gg k$. The solutions involve Bessel functions whose order in the $y$-sector depends parametrically on the $z$-sector quantum number $p$, leading to a double-indexed mass spectrum.

The resulting KK tower derives from nested transcendental boundary conditions: first, $m_p$ is found via

$$5 J_{\nu_\phi}(x_p) + x_p J_{\nu_\phi}'(x_p) = 0 \quad \text{with} \quad x_p = (m_p/k')e^{-k\pi}$$

Subsequently, for each $m_p$, one obtains the full 4D KK mass $m_{np}$ from: $$7 J_{\nu_p}(x_{np}) + x_{np} J_{\nu_p}'(x_{np}) = 0, \quad x_{np}=m_{np} e^{c\pi}/k'$$ The generically split double-tower spectrum is a hallmark of multiply warped models, distinguishing them from the canonical 5D Randall-Sundrum scenario (Koley et al., 2010).

3. Mass Spectrum, Mode Splitting, and Sterility

The mass spectrum in these backgrounds displays clustering: for each $p$ (from the $z$-dimension), a sub-tower of modes labeled by $n$ (from the $y$-dimension) emerges, and the order of the Bessel function ($\nu_p$) depends on $p$. This dependency causes the appearance of double branches (or “splitting”) in the TeV mass range, with the detailed structure sensitive to quantization in both extra dimensions.

The sterile nature of KK states is reflected in their wavefunction localization—the vast majority of modes have negligible overlap with the brane or Standard Model fields. Only the lowest (typically the $n=p=0$) mode may be strongly coupled to visible sector physics, while higher modes typically couple via Planck-suppressed or warping-induced exponentially small coefficients. The sterile KK states thus populate the spectrum densely but remain effectively undetectable except through their gravitational or higher-order effects (Koley et al., 2010).

4. Phenomenological Implications and Collider Signatures

Clusters of sterile KK states in the TeV region modify collider signatures in several ways:

• Resonances in scattering amplitudes can arise due to large degeneracies or near-degeneracies in the mass spectrum, possibly leading to enhancements in processes involving missing energy or invisible decays.
• Modification of decay rates, scattering amplitudes, and cross sections due to virtual sterile KK exchanges, with contributions stronger than in 5D models owing to the increased mode density.
• The possibility of indirectly inferring sterile KK towers by searching for anomalous event rates or resonance patterns at energies corresponding to the cluster spacing predicted by the nested transcendental equations.

The detectability enhances with the number of warped dimensions, as each introduces additional spectral clustering near the electroweak scale. This property enables experimental discrimination between models with different numbers of warped extra dimensions (Koley et al., 2010).

5. Comparison with Minimal Randall–Sundrum and Larger Dimensions

In the original 5D Randall–Sundrum model, the KK spectrum is characterized by a single tower associated with fixed-order Bessel functions, typically separated by mass gaps much larger than those resulting from extra splittings in multiply warped scenarios. The introduction of multiple warped directions, and consequently multiple sets of quantum numbers, yields exponentially larger numbers of sterile modes at accessible energies. Generalization to $D>6$ dimensions—i.e., $M^{1,3}\times[S^1/\mathbb{Z}_2]^{D-4}$—exponentially increases the proliferation of sterile states and the complexity of the spectrum, with each additional warped dimension effectively introducing another layer of mass clustering (Koley et al., 2010).

6. Theoretical Extensions and Outlook

The formalism and findings naturally generalize to scalar, vector, and higher-spin fields, as well as to more complicated topologies with larger numbers of orbifolded, warped extra dimensions. Extension to higher $D$ dimensions results in a combinatorial growth in the number of sub-towers and further enhancement in sterile KK mode multiplicity. Such proliferation offers more stringent phenomenological tests due to the richer collider implications and further opportunity for distinguishing extra-dimensional models in upcoming experiments (Koley et al., 2010).

The theoretical structure ensures that the mass splitting and clusters are a generic property of multiply warped compactifications, not artifacts of specific parameter choices. Furthermore, the nesting of mass eigenvalue equations provides a systematic, model-independent route for calculating physical KK spectra, enabling rigorous predictions for sterile KK states across braneworld settings with multiply warped internal spaces.

7. Summary Table: Key Features in Multiply Warped Sterile KK Spectroscopy

Aspect	Multiply Warped Model	Randall-Sundrum (5D)
KK mass spectrum	Double-indexed (split towers), clustered	Single-indexed (simple tower)
Mode eigenfunctions	Products of Bessel functions (order depends on both $y$ and $z$)	Bessel functions (fixed order)
Phenomenology	Enhanced number, TeV clusters, richer collider potential	Fewer modes, wider spacing
Sterility origin	Weak brane coupling, wavefunction localization	Weak brane coupling, less mode density
Extension to $D>6$	Multiple nested clustering, combinatorial proliferation	No direct analog

This structured landscape—arising from multiply warped geometries—renders the identification and theoretical analysis of sterile Kaluza-Klein states an essential component of modern extra-dimensional model building and phenomenological exploration.