Gapped Unparticles

• Gapped unparticles are continuous excitation spectra emerging above a mass gap, breaking strict scale invariance with a thresholded spectral density.
• They are realized via IR deformations, extra-dimensional soft-wall models, and holographic duality, with analytic control offered by dispersive two-point functions.
• Applications range from modifying collider observables to explaining pseudogap phenomena in strongly correlated systems, providing predictive theoretical frameworks.

Gapped unparticles are excitations forming a continuous spectrum above a nonzero threshold (mass gap), realized in quantum field theory and phenomenology as deformations of scale-invariant unparticle sectors. In contrast to traditional unparticles—whose scale invariance leads to continuous spectra starting from zero invariant mass—gapped unparticles acquire an explicit lower threshold for the onset of the continuum, either via relevant deformations, compactification, or explicit breaking of conformal symmetry. This structure has been explored both as a calculable theoretical tool and as a predictive framework for strongly coupled sectors in collider and condensed matter physics, as well as in model-building employing extra dimensions and holographic duality.

1. Fundamental Structure and Defining Features

The central property of gapped unparticles is the presence of a continuum of states with a spectral density that is strictly zero below a threshold $m_\mathrm{gap}$, and exhibits power-law scaling characteristic of conformal dynamics above this gap. The canonical two-point function admits a dispersive (Källén-Lehmann) representation: $$\langle \mathcal{O}(p)\,\mathcal{O}(-p)\rangle = \int_{m_\mathrm{gap}^2}^{\infty}\!\!ds\,\frac{\rho(s)}{p^2-s+i\epsilon},$$ with

$$\rho(s) = \theta(s - m_\mathrm{gap}^2)\,A_U\,(s - m_\mathrm{gap}^2)^{d_U-2}.$$

Here, $d_U$ is the scaling dimension of the unparticle operator and $A_U$ is a normalization constant. Introducing the gap amounts to replacing the strictly scale-invariant spectral density $\sim s^{d_U-2}$ (which would extend down to $s=0$) with a thresholded form, $\sim (s - m_\mathrm{gap}^2)^{d_U-2}$ (Arganda et al., 2024).

The propagator in momentum space develops a branch cut starting at $p^2 = m_\mathrm{gap}^2$, with the branch point singularity governed by $d_U$, leading to the characteristic "gapped unparticle" response. In spacetime, the two-point function exhibits power-law scaling at short/intermediate distances and exponential decay at separations larger than $1/m_\mathrm{gap}$ (Georgi et al., 2019).

2. Theoretical Realizations and Holographic Duality

Gapped unparticles are realized in several frameworks, notably:

• 4D Field Theory with IR Deformations: Starting from Georgi's unparticle sector, a mass gap can be introduced by IR-relevant perturbations or explicit mass terms, yielding the gap-thresholded spectral density above [(Arganda et al., 2024); (Phillips, 2014)].
• Extra Dimensional Models with Soft-Wall Geometries: In warped 5D models, a gap arises naturally when the space ends at a curvature singularity ("soft wall") or due to dilaton backgrounds, as in models by Megías–Quirós (Megias et al., 2019) and Cai–Cheng–Medina–Terning (Cheng, 2010). The metric is typically

$$ds^2 = e^{-2A(y)}\,\eta_{\mu\nu}dx^\mu dx^\nu - dy^2,$$

where the IR structure of $A(y)$ dictates the emergence and size of the continuum gap. Fluctuation equations for bulk fields reduce to Schrödinger-type problems with potentials $V(z)\to V_\infty>0$ at large $z$, resulting in a continuum spectrum for $p^2 > V_\infty$ and a mass gap $m_\mathrm{gap}^2 = V_\infty$ (Megias et al., 2019).

• Holographic Duals of Gapped Gravity and Generalized Dimensional Reduction: In gapped AdS gravity, the spectrum of the dual energy-momentum tensor two-point function displays non-integer power-law scaling and a mass threshold set by the Fierz–Pauli mass of the bulk graviton (Domokos et al., 2015). An alternative construction involves starting from a higher-dimensional conformal field theory and compactifying one or more dimensions, yielding a tower of "gapped unparticle" modes, a hybrid of discrete gaps from Kaluza-Klein quantization and non-integer scaling inherited from the parent CFT (Jiang et al., 29 Dec 2025).
• Toy Models via First-Quantized Actions: First-quantized models such as that of Casalbuoni–Gomis and its deformations yield analytic control over transitions between purely conformal, gapped unparticle, and confining regimes (Boulanger et al., 2021). The tensionless limit of confining potentials reproduces a gapped continuous spectrum.

3. Spectral Properties, Green's Functions, and Operator Dynamics

The operator content and correlation structure of gapped unparticles is governed by three distinct regimes:

1. Short distances ($x \ll 1/M$): correlation functions mirror those of the underlying UV CFT or free-fermion behaviors.
2. Intermediate regime ($1/M \ll x \ll 1/m_\mathrm{gap}$): unparticle power-law scaling, $\langle O(x)O(0)\rangle \sim |x|^{-2\gamma}$, with $\gamma$ the anomalous dimension appropriate to the operator and gauge sector (Georgi et al., 2019).
3. Long distances ($x \gg 1/m_\mathrm{gap}$): exponential decay, $$\langle O(x)O(0)\rangle \sim e^{-m_\mathrm{gap}|x|}$$.

The two-point function and its spectral density encode a hard threshold at $p^2 = m_\mathrm{gap}^2$. Standard Model fields propagating in soft-wall backgrounds have field-dependent $m_\mathrm{gap}$ set by the asymptotic form of effective 1D potentials. For example, for a gauge boson $V_A \to \frac{1}{4}\rho^2$, leading to $m_\mathrm{gap} = \rho/2$ (Megias et al., 2019).

Cluster decomposition and operator coalescence imply that all operators of the same quantum number coalesce into a unique gapped unparticle operator in the IR (Georgi et al., 2019). In compactified higher-dimensional setups, the correlator is a sum over all tower modes, each contributing as a "gapped unparticle" with its own threshold and envelope controlled by the (generally non-integer) parent scaling dimension (Jiang et al., 29 Dec 2025).

4. Phenomenological Implications and Collider Constraints

Gapped unparticles alter the dynamics of high-energy processes in fundamental ways:

• Deviations in Scattering Observables: In soft-wall 5D models, the exchange of Standard Model gauge bosons propagating in the extra dimension results in amplitudes with the typical SM pole plus a continuum branch cut above $m_\mathrm{gap}$. This structure leads to modified Drell–Yan and monolepton invariant mass spectra at hadron colliders, as continuum contributions interfere with the SM resonance amplitudes (Arganda et al., 2024).
• Experimental Bounds: Analyses of dilepton and monolepton events at the LHC, incorporating full matrix-element reweighting and event simulation, yield lower bounds on the continuum scale parameter $\rho$:
  • Expected: $\rho > 4.2$ TeV
  • Observed: $\rho > 6.2$ TeV (merging bins with $<10$ events)
  • Projected HL-LHC: $6.5$–$8.1$ TeV depending on systematic assumptions (Arganda et al., 2024).
• Spectral Response: The presence of a hard threshold means that high-mass excesses in collider observables are distributed smoothly above the gap rather than appearing as sharp resonance peaks. LHC searches for continuum superpartners or other soft-wall extensions must focus on broad excesses and altered tails.
• Implications for Extended Sectors: In supersymmetric soft-wall constructions, gapped unparticle spectra realize "continuum superpartners," where would-be SUSY partners form a continuum above a gap, drastically modifying cascade decays and the signatures of missing energy at colliders (Cheng, 2010).

5. Connections to Strongly Correlated Matter and Nonlocality

Gapped unparticle sectors are directly relevant to non-Fermi liquid states and the breakdown of quasiparticle descriptions in strongly correlated electron systems:

• Spectral Zeros and Pseudogaps: In $\mathrm{cuprate}$ pseudogap phases, the unparticle formalism yields a single-particle propagator with zeros at the chemical potential, reflecting a suppression of spectral weight without symmetry breaking [(Phillips et al., 2013); (Phillips, 2014)]. The introduction of a hard gap further suppresses low-energy excitations, and the anomalous scaling dimension is fixed by the underlying bulk dual or continuous-mass construction.
• Superconducting Instabilities: The unparticle BCS gap equation, when adapted to gapped unparticle propagators, leads to the unconventional result that the superconducting transition temperature increases as the attractive interaction decreases—the system is "super-susceptible" to pairing for small couplings [(Phillips, 2014); (Phillips et al., 2013)].
• Nonlocality and Scaling: Holographic duals of gapped unparticles are intrinsically nonlocal, with two-point functions scaling as $p^{2\tilde\Delta}$, where $\tilde\Delta$ is set by the bulk mass, rather than the integer power expected from local CFTs (Domokos et al., 2015).

6. Model Catalog and Gap Values Across Sectors

A summary of characteristic gap values and spectra for key scenarios:

Field Type	$V_\infty$ (as $z\to\infty$)	$m_\mathrm{gap}$	Comments
Gauge boson	$\frac{1}{4}\rho^2$	$\rho/2$	Smallest gap, $O(\mathrm{TeV})$
Fermion ($c_\psi$)	$(c_\psi \rho)^2$	$|c_\psi|\rho$	Gaps depend on localization
Graviton, Radion	$(\frac{3}{2})^2\rho^2$	$3\rho/2$	Universal for bulk scalars
Higgs fluctuation	$(\frac{3}{2})^2\rho^2$	$3\rho/2$	Isolated pole at $m_H$

In 5D models, the bosonic continuum of KK modes with the smallest mass gaps are those of gauge bosons, making them the most likely to be produced at the LHC. Mass gaps for fermionic continua depend sensitively on their localization in the extra dimension (via $c_\psi$). These thresholds, spectral functions, and Green's functions determine modifications to Standard Model processes (Megias et al., 2019).

7. Analytical Control, Toy Models, and Generalizations

First-quantized models furnish analytical interpolation between the gapped unparticle regime and confining phases. Adding a linear potential to the conformal Casalbuoni–Gomis model produces Regge trajectories, while the tensionless limit yields a continuous spectrum with finite mass gap. For low-lying states, conformal restoration eliminates the mass gap and yields massless bound states, maintaining dimensionless ratios as $\sigma \to 0$ (Boulanger et al., 2021).

In higher-dimensional and inflationary contexts, compactification yields a tower of gapped unparticle modes whose exchanges contribute oscillatory signatures in cosmological correlation functions. The envelope exponent is governed by the anomalous scaling dimension inherited from the parent CFT, offering discriminability from ordinary massive fields (Jiang et al., 29 Dec 2025).

• Drell-Yan Bounds on Gapped Continuum Spectra (Arganda et al., 2024)
• Unparticles as the Holographic Dual of Gapped AdS Gravity (Domokos et al., 2015)
• Gapped Continuum Kaluza-Klein spectrum (Megias et al., 2019)
• Continuum Superpartners (Cheng, 2010)
• Beyond Particles: Unparticles in Strongly Correlated Electron Matter (Phillips, 2014)
• Non-perturbative Effects and Unparticle Physics in Generalized Schwinger Models (Georgi et al., 2019)
• Un-Fermi Liquids: Unparticles in Strongly Correlated Electron Matter (Phillips et al., 2013)
• Strongly Coupled Sectors in Inflation: Gapped Theories of Unparticles (Jiang et al., 29 Dec 2025)
• A First-Quantized Model For Unparticles and Gauge Theories Around Conformal Window (Boulanger et al., 2021)
• Holographic Lieb lattice and gapping its Dirac band (Han et al., 2022)