# Dark Photons and Their Correlation with The Dark Sector Gravitons and The Higgs Boson $H \to \gamma\gamma$ Decay

Emmanouil Markoulakis, Antonios Valamontes

► To cite this version:

Emmanouil Markoulakis, Antonios Valamontes. Dark Photons and Their Correlation with The Dark Sector Gravitons and The Higgs Boson $H \to \gamma\gamma$ Decay. International Journal of Physical Research, 2025, 13 (2), pp.1 - 13. 10.14419/xhvxrg97. hal-05204263v2

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Submitted on 14 Oct 2025

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# Dark Photons and Their Correlation with The Dark Sector Gravitons and The Higgs Boson H→γγ Decay

Emmanouil Markoulakis<fnref n="1" />*, Antonios Valamontes<fnref n="2" />

Received: June 4, 2025, Accepted: June 29, 2025, Published: August 7, 2025

## Abstract

In this paper, we present an in-depth phenomenological and technical description of the ATLAS LHC experiment, focusing on two-γ-photon decay events of the Higgs boson. We explore the implications of these decay events in the context of our novel theory, the Superluminal Graviton Condensate Vacuum (SGCV). This theory predicts the existence of "superluminal propagating massless dark photons", which act as ghost particles of superluminal displaced vacuum dark sector gravitons. The detection of these particles would establish a significant link between the Higgs field, the quantum field of normal photons, and the dark sector. We propose experimental setups and Time-of-Flight (ToF) instrumentation detectors to validate these predictions, emphasizing their importance for understanding the vacuum and the fabric of spacetime, which is hypothesized to be fundamentally composed of superluminal vibration energy.

**Keywords:** Quantum Gravity; Zero-Point Vacuum Energy; Graviton; Higgs Boson; Dark Photons; Time-of-Flight (ToF) Detector; Dark Sector Experiment; Quantum Cosmology.

## 1. Introduction

The authors' previously published research papers [1] [2] conclude that superluminal rest energy is essential<fnref n="1" /> for our whole universe to exist. Therefore, also essential for the formulation of a theory of Quantum Gravity (QG) that the vacuum and Dark Energy (DE) are intrinsically hidden superluminal energy.

In the past, many papers have been written [3–9] [10–20] about the topic of the postulated dark photons and their detection experiment attempts, which have proved so far all unfruitful. The authors make a bold ansatz herein that the whole dark sector may be intrinsically at the micro-Planck scale and beyond the Planck Frequency<fnref n="2" /> of energy vibrations, superluminal energy. An idea supported theoretically also by quantum field theory QFT and quantum electrodynamics QED when the Ultraviolet Divergence calculation results are taken seriously into consideration.

One way to validate this theory [1] is via the detection of the generally theorized dark photons but with the twist the authors introduce that these could also be superluminal propagating in the vacuum, thus faster than light c speed limit of normal photons without violating special relativity, since these reside in the dark sector and probably unknown new physics. Next, we will discuss the phenomenology and predictions of this theory and how it can be tested by experiment. Also, for the detailed derivation of all the equations presented inside the paper, follow the “supplementary material” link at the end of the paper.

## 2. Results and discussion

### 2.1. Phenomenology of theory and experiment validation

We propose a novel LHC Atlas experiment using Time-of-Flight (ToF) (i.e., hardware and s/w need modifications to adapt to this experiment) measurements of the collision decay byproducts searching for signature signals of superluminal propagation of particles inside the vacuum. This experiment proposal can also be implemented and possibly more easily in the CERN SHiP³ detector project. The GeV Beam range output of the SHiP detector hitting a sample target material should be enough to produce possible "Superluminal Dark Photons" with superluminal ToF inside the vacuum (i.e., faster than light c speed, FTL), the authors' novel quantum theory [1] predicts that they exist.

Specifically, according to this theory [1] [2] dark photons are something different than initially believed and namely they are ghost image imprints of displaced vacuum gravitons [1] (like a cloud chamber particle trail) and only the dark photons that are propagating superluminal inside the vacuum (larger than c speed) can be indirectly detected. This is because normal c speed dark photons would quickly

<footnote n="1">¹ Department of Electronic Engineering, Hellenic Mediterranean University, Romanou 3, Chania 73133, Crete, Greece</footnote>

<footnote n="2">² University of Maryland, Munich Campus, Tegernseer Landstraße 210, 81549 München, Germany and Kapodistrian Science Foundation, Largo, 33774, Florida, USA</footnote>

<footnote>*Corresponding author E-mail: markoul@hmu.gr</footnote>
diffuse inside the zero-point-energy (ZPE) field of the vacuum, its zero-point-fluctuations (ZPF), and would be very hard to impossible to isolate from the background noise of the vacuum.

In section 2.7 of the previous [1] paper, we outlined an LHC Atlas experiment to indirectly verify the theory of the "superluminal rest energy vacuum", by detection of superluminal propagating dark photons during a Higgs decay event to two normal γ-photon by measuring the:

H → γγ flight path distances and Time of Flight (ToF) to the Atlas detector's L(Ar) EM calorimeter sensors.

Dark photons would effectively make the normal photons (after these dark photons turned into normal photons) initially appear in a displaced position from the Atlas detector's chamber center, and assuming dark photons are the ghost particles of the superluminal displaced gravitons [1] of a "Superluminal Graviton Condensate Vacuum" (SGCV).

Notes: ¹The law of conservation of energy and the second law of thermodynamics do not hold at the cosmological scale if superluminal rest energy of the vacuum is not included in a theory.

$$ ^2Fp = \frac{1}{2\pi} \sqrt{\frac{c^5}{hG}} = Ep/h \approx 2.952 \cdot 10^{42} \text{ Hz} $$

³https://ship.web.cern.ch/

This would then make the initial two superluminal propagating dark photons produced and then turned into the two observed in the Atlas detector normal γ-photon produced in a Higgs Boson decay event, and these normal γ-photon traveling at luminal normal speed c. But now, because their path length to the sensors has shortened due to the initially produced superluminal travelling dark photons then this would make them effectively appear as if they have traveled the total distance from the center of the detector chamber at superluminal speed.

The authors postulate that the graviton is entangled with its ghost particle, the dark photon [1].

This author's theory that dark photons are actually "ghost image imprints" or simply called ghost particles of nearfield superluminal displaced cold gravitons [1] inside the Superluminal Graviton Condensate Vacuum (SGCV), can be visualized with the following analogy example and thought experiment.

## 2.2. The graviton-bullet in a water pool analogy and thought experiment

Consider you shoot a supersonic bullet on a pool water surface and assume that this bullet, for the first meter inside the water, travels faster than the speed of sound in water. What will the water experience as the bullet slices through the water mass? The water (here analog to the ZPE vacuum) will be distorted and displaced, forming lines and whirls (here analog to EM flux and particles, namely in our case normal photons) that propagate at the speed of sound in water, thus about, although the bullet propagates much faster at supersonic speed for the first meter of penetration.

The authors' novel prediction according to their research [1] about the dark photons connection with normal photons and how it is possible to have superluminal propagation of EM signal information for the near-field frequencies: The bullet splashes on the pool water surface at point A and although the water distortion travels along the longitudinal trajectory path of the bullet with speed no more than the speed of sound in water, a distortion of the water appears at supersonic time at point B, one meter below the water surface. This ansatz is illustrated in Fig. 1 (a) & (b).

<figure><img src="image_2.png" /><figcaption>Fig. 1: A) Illustration Describing How You Can Have A Near-Field Observed Superluminal Propagation of EM Waves (Photons) without Exceeding the Absolute Speed of Light C in the Vacuum B) the Analogy Example with A Supersonic Bullet: the Graviton Thus the Bullet in Analogy In Our Example, Is Displaced Superluminal (Supersonic in Analogy with the Bullet) Creating A Superluminal Propagating Trail of Dark Photons While at the Trailing End of the Graviton (Bullet) Creates Normal Photons Lorentz-Invariant Propagating at Luminal Speed C (The Speed of Sound in Air Or Water in Our Bullet Analogy Example).</figcaption></figure>

Source image credits: SmarterEveryDay https://www.youtube.com/watch?v=BPwdlEgLn5Q.

So the question now becomes, what is this bullet in our case? According to our research [1], the actual "vacuum molecule" consisting of everything and the only elementary particle there is, is the sub-Planck-sized superluminal in-place vibration cold graviton, which can, however, be displaced even superluminal from its initial rest position (for example, during a LHC beams collision). The supersonic bullet in our example is the superluminal displaced graviton during the Higgs Boson LHC Atlas experiment.
Notice here, most importantly, that in our theory [1], the Higgs Boson is just a static scalar imprint of a decelerated to c vibration speed source superluminal vibrating graviton of the vacuum during a beam collision LHC experiment. Meaning, a Higgs spin-0 Boson is fundamentally the ghost particle of a vacuum spin-2 superluminal graviton Boson.

Coming back to our thought experiment and analogy example, the front trail the bullet leaves slicing through the water are the superluminal propagating dark photons thus ghost image of the superluminal displaced graviton (see Fig. 1b) during its motion and the splashes distortions in point A and point B are the two normal γ- photons created we see in an LHC Higgs Boson $H \to \gamma\gamma$ decay event, see Fig. 2.

However, the distortion of the γ-photon in point B would effectively appear as if it had traveled at superluminal time from point A.

Similarly, the two γ-photon observed in a Higgs Boson decay event in LHC Atlas, see Fig. 2, would appear to have a superluminal Time-of-Flight (ToF) from the beams' collision point at the center of the Atlas chamber to the outer perimeter calorimeter sensors.

<figure><img src="image_3.png" /><figcaption>Fig. 2: Higgs Boson of Spin-0 Being Actually a Static Scalar Imprint (I.E., Ghost Particle) of the Superluminal Spin-2 Graviton [1].</figcaption></figure>

However, to perform this proposed experiment at the Atlas detector, this detector needs some hardware and s/w upgrades since it is currently not equipped [21] at its current state, as shown in Fig. 3, to execute such Time-of-Flight (ToF) measurements we describe herein.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Illustration of LHC Atlas Detector Inner Core [21].</figcaption></figure>

This Time-of-Flight (ToF) measurement could be relative easily performed by identifying each time the trajectory path length of the decay product photons [22] and measuring their propagation time from $\Delta t=0$ of the collision event from the center of the detector chamber to the calorimeter sensors as illustrated in

<figure><img src="image_5.png" /><figcaption>Fig. 4: Tof Measurement of A Specific Trail Length of the Two Γ Photons Produced in A $H \to \gamma\gamma$ Decay Event in LHC Atlas. Notice, the Shortest perpendicular to the Calorimeters Trajectory Path Is Approximately 1.04 M in Length (Not Indicated in the Above Illustration) [22].</figcaption></figure>

Possibly, the new SHiP CERN detector and project is currently the most suitably equipped for this proposed herein experiment and can easily modified to include also ToF measurements for superluminal signals created by dark photons as predicted by the authors [1].
## 2.3. Analytical framework of the SGCV model and measurements

Understanding the fabric of spacetime and the vacuum structure has been a central pursuit in theoretical physics. The Superluminal Graviton Condensate Vacuum (SGCV) model [1] suggests that gravitons form a chaotic, ideal fluid-like condensate that defines the vacuum itself.

**Boundary Conditions:**

For the small-scale chaotic regime governed by the SGCV model, the vacuum can be considered homogeneous and isotropic at very small distances, so the boundary conditions are,

$$ \lim_{r \to 0} \psi_{SGCV}(x, t) = \psi_0, \quad \lim_{r \to \infty} \psi_{SGCV}(x, t) = 0 \qquad (1) $$

This ensures that the wavefunction is well-behaved at small distances and decays to zero at infinity, consistent with a localized graviton condensate.

**Symmetry Considerations:**

At small scales, the SGCV symmetry is highly chaotic, and we assume an isotropic, ideal fluid-like vacuum. The underlying symmetry group is $SO(3)$, corresponding to rotational invariance. The fluid-like vacuum does not distinguish between directions.

**Energy Density in SGCV:**

At small scales, the vacuum energy density is dominated by chaotic fluctuations, modeled as:

$$ \rho_{SGCV}(x) = \rho_0 + \delta\rho(x) \qquad (2) $$

Where $\rho_0$ is the average energy density and $\delta\rho(\chi)$ represents local fluctuations due to quantum uncertainty.

The energy density fluctuations $\delta\rho(\chi)$ follow a power spectral density $P(k)$ that describes how energy is distributed across different wavelengths:

$$ P_{SGCV}(k) \propto k^2 e^{-k/k_0} \qquad (3) $$

Where $k_0$ is the characteristic wavenumber related to the correlation length of graviton fluctuations.

**Renormalization and Scale Dependence of Coupling Constants:**

As the vacuum structure changes across scales, the physical parameters (such as interaction potentials and coupling constants) also evolve. We apply renormalization group techniques to understand how these quantities change with the energy scale $\lambda$.

The coupling constant $\lambda_{SGCV}$, which controls the interaction strengths, evolves according to the renormalization group equations (RGEs).

In general, the renormalization of the coupling constants is given by,

$$ \frac{d\lambda_{SGCV}}{d \log \mu} = \beta_{SGCV}(\lambda_{SGCV}) \qquad (4) $$

Where $\mu$ is the energy scale and $\beta_{SGCV}$, is the beta function. These equations (4) describe how the coupling constants run with the energy scale, ensuring a smooth transition between different size scales.

**Higher-Dimensional Considerations:**

The model often assumes a higher-dimensional vacuum structure, particularly in the context of superluminal quanta. In this theory, we include higher-dimensional effects by extending the vacuum structure to $D>3$ spatial dimensions.

**Dimensional Transition:**

At small scales, the effective dimensionality $D$ is close to three due to the chaotic nature of the vacuum. Still, at larger scales, the dimensionality may increase due to the higher-dimensional embedding.

We model the dimensionality $D(\lambda)$ as a scale-dependent function:

$$ D(\lambda) = 3 + (D_{max}(1 - f(\lambda))) \qquad (5) $$

Where $D_{max}$ is the maximum dimensionality of the vacuum in the macro scale regime. This smoothly interpolates between 3-dimensional chaos (SGCV) and higher-dimensional structure.

**Compaction:**

The extra dimensions are compactified at small scales. The compactification radius $R_c(\lambda)$ is scale-dependent and decreases with increasing energy,

$$ R_c(\lambda) \sim \frac{1}{\lambda^\gamma} \qquad (6) $$

Where $\gamma>0$ determines how quickly the compactification occurs as the vacuum transitions from the chaotic to the structured phase.

**Graviton Interaction Potentials:**

The interaction potentials between gravitons are central to understanding the vacuum structure. At small scales, the interaction is dominated by short-range forces, while at large scales, long-range lattice interactions become relevant.

**SGCV Graviton Interaction Potential:**

At small scales, the graviton interaction is chaotic and dominated by local fluctuations. The potential can be modeled as:

$$ V(r) = \int \lambda_{SGCV} r^2 dr = \frac{\lambda_{SGCV} r^3}{3} + C \qquad (7) $$
Where $\lambda_{SGCV}$ captures short-range repulsive forces. The $r^2$ term represents the chaotic fluctuations that dominate at small distances, and C is the integration constant. For an in-depth analysis and derivations of the above, see Appendix II, section 2.3.4 in the supplementary material.

The experimental proposal to detect superluminal dark photons relies on Time-of-Flight (ToF) measurements at CERN's Atlas detector and the SHiP project. To estimate the feasibility of these measurements, we perform an error analysis on the detection sensitivity.

Sensitivity of ToF Measurements:

To detect superluminal dark photons, the ToF measurement must have a resolution $\Delta t$ capable of distinguishing between luminal and superluminal particles. The minimum time difference detectable is given by:

$$ \Delta t_{\min} = \frac{L}{c} \left( 1 - \frac{v_{\text{dark}}}{c} \right) \qquad (8) $$

Where L is the path length, $v_{\text{dark}}$ is the speed of the dark photon, and c is the speed of light.

Assuming that $v_{\text{dark}} \approx 1.01c$, we require a timing resolution on the order of picoseconds for accurate detection.

The primary sources of noise include background radiation, thermal noise in the detector, and uncertainty in the photon paths. These sources contribute to the overall noise level $N_{\text{total}}$, which must be minimized to achieve sufficient sensitivity:

$$ \text{SNR} = \frac{s}{N_{\text{total}}} \qquad (9) $$

Where S is the signal strength and $N_{\text{total}}$ is the total noise level. For detection to be feasible, we need SNR>5.

Monte Carlo simulations are employed to model the transition between chaotic and micro-scale structured regimes and predict observable quantities such as graviton correlations, energy fluctuations, and cross-sections for dark photon interactions. The detailed results of the numerical analysis are described in Section 2.4 and Appendix II in the supplementary material link at the end of the paper.

The graviton correlation function is computed using stochastic methods, and it evolves as:

$$ C(r) = f(\lambda)C_{\text{SGCV}}(r) + (1 - f(\lambda))C_{\text{DLSFH}}(r) \qquad (10) $$

Where $C_{\text{SGCV}}(r) \sim e^{-r/\xi_{\text{SGCV}}}$.

The cross-section for superluminal dark photon production during Higgs decay is predicted to be,

$$ \sigma_{\text{dark}}(s) = \frac{\alpha^2}{s} \left( \log \left( \frac{s}{m_{\text{dark}}^2 c^2} \right) \right) \qquad (11) $$

Where S is the center-of-mass energy, and $m_{\text{dark}}$ is the effective mass of the dark photon.

While the unified framework presented here offers a coherent picture of the vacuum structure, it is not without limitations:

* Model Validity: The transition between the SGCV and DLSFH models is phenomenological, and a more rigorous derivation from first principles may be needed.

* Experimental Challenges: Detecting superluminal particles requires unprecedented precision in ToF measurements, and the results could be contaminated by background noise.

* Numerical Complexity: The Monte Carlo simulations require significant computational resources, and convergence might be slow in certain regimes.

* Using the scale parameter $\lambda$ and the transition function $f(\lambda)$, the hybrid vacuum model is constructed by combining the contributions from both the SGCV [1] and DLSFH models [23]. The total vacuum tensor field $v_{\mu\nu}(\lambda, \chi)$ is defined as a weighted sum of the SGCV and DLSFH components:

$$ V_{\mu\nu}(\lambda, x) = f(\lambda)V_{\mu\nu}^{\text{SGCV}}(x) + (1 - f(\lambda))V_{\mu\nu}^{\text{DLSFH}}(x) \qquad (12) $$

Where:

* $V_{\mu\nu}^{\text{SGCV}}(x)$ represents the chaotic, fluid-like vacuum structure at small scales.

* $V_{\mu\nu}^{\text{DLSFH}}(x)$ represents the structured, lattice-like vacuum structure at large macro scales.

For $\lambda \ll \lambda_c$, the transition function $f(\lambda)$ approaches 1, and the total vacuum tensor field behaves predominantly according to the SGCV model. As $\lambda$ increases beyond $\lambda_c$, $f(\lambda)$ decreases and the contribution from the macro DLSFH model becomes more dominant.

Transition of Graviton Interaction Potential:

The graviton interaction potential also evolves with the scale parameter $\lambda$. The potential, $V(r)$, is a hybrid of the SGCV and DLSFH interaction potentials, weighted by the transition function $f(\lambda)$:

$$ V(\lambda, r) = f(\lambda)V_{\text{SGCV}}(r) + (1 - f(\lambda))V_{\text{DLSFH}}(r) \qquad (13) $$

Where,

$V_{\text{SGCV}}(r) = -\frac{G}{r} + \lambda_{\text{SGCV}} \frac{r^2}{2}$ is the chaotic interaction potential, which dominates at small distances r, typical of the SGCV regime.

$V_{\text{DLSFH}}(r) = -\frac{G}{r} + \lambda_{\text{DLSFH}} \cos(kr)$ is the structured lattice interaction potential, which dominates at larger distances, typical of the macro DLSFH regime.

The transition function $f(\lambda)$ controls how the vacuum interaction potential evolves with scale.

At small scales, where $\lambda \ll \lambda_c$, the SGCV interaction potential governs the short-range behavior of gravitons. As the scale increases beyond $\lambda_c$, the macro DLSFH interaction potential becomes dominant, introducing long-range correlations and lattice-like behavior.
Scale-Dependent Wavefunction of the Graviton Condensate:

$$i\hbar \frac{\partial \psi(\lambda, x)}{\partial t} = \left( -\frac{\hbar^2}{2m_g} \nabla^2 + V(\lambda, x) \right) \psi(\lambda, x) \qquad (14)$$

The graviton condensate wavefunction, $\psi(\lambda, \chi)$, evolves according to a scale-dependent Gross-Pitaevskii equation. The dynamics of $\psi(\lambda, \chi)$ are governed by the effective interaction potential, which depends on the energy scale $\lambda$. Here, $V(\lambda, \chi)$, is the hybrid interaction potential, which evolves as:

$$V(\lambda, x) = f(\lambda)V_{\text{SGCV}}(x) + (1 - f(\lambda))V_{\text{DLSFH}}(x) \qquad (15)$$

At small scales ($\lambda \ll \lambda_c$), the wavefunction evolves in a chaotic, fluid-like manner. As $\lambda$ increases, the wavefunction transitions to a more structured state, reflecting the underlying lattice of the DLSFH model. Wavefunction of the Graviton Condensate:

The wavefunction $\psi(\lambda, \chi)$, describes the quantum state of the graviton condensate, evolving under the influence of a scale-dependent interaction potential. This follows a Gross-Pitaevskii equation with self-interaction terms included:

$$i\hbar \frac{\partial \psi(\lambda, x)}{\partial t} = \left( -\frac{\hbar^2}{2m_g} \nabla^2 + V(\lambda, x) + g_{\text{self}} |\psi(\lambda, x)|^2 \right) \psi(\lambda, x) \qquad (16)$$

Where,

* $V(\lambda, \chi)$ is the hybrid potential.

* $g_{\text{self}}$ represents the strength of graviton self-interactions.

* The self-interaction term $g_{\text{self}} |\psi(\lambda, x)|^2$ accounts for the interactions between gravitons within the condensate.

Normalization of the Wavefunction:

The wavefunction must be normalized to ensure physical consistency. The total probability of finding a graviton in the vacuum is conserved:

$$\int |\psi(\lambda, x)|^2 d^3 x = 1 \qquad (17)$$

This normalization condition holds across both chaotic and structured regimes, ensuring that the wavefunction remains physically meaningful as the vacuum transitions between SGCV and DLSFH behavior.

Stress-Energy Tensor of the Vacuum:

The stress-energy tensor $T_{\mu\nu}$ captures the energy density and pressure of the vacuum, linking the vacuum dynamics to general relativity. The stress-energy tensor is derived from the vacuum action $S_{\text{vacuum}}$,

$$T_{\mu\nu} = \frac{\delta S_{\text{vacuum}}}{\delta g^{\mu\nu}} \qquad (18)$$

The total stress-energy tensor can be decomposed into contributions from the SGCV and macro DLSFH components:

$$T_{\mu\nu} = f(\lambda)T_{\mu\nu}^{\text{SGCV}} + (1 - f(\lambda))T_{\mu\nu}^{\text{DLSFH}} \qquad (19)$$

Where $T_{\mu\nu}^{\text{SGCV}}$ and $T_{\mu\nu}^{\text{DLSFH}}$ describe the stress-energy contributions from the chaotic and structured vacuum components, respectively. These components influence the gravitational behavior of the vacuum, potentially contributing to effects such as vacuum energy and spacetime curvature.

Renormalization Group Flow for Coupling Constants:

The coupling constants $\lambda_{\text{SGCV}}$ and $\lambda_{\text{DLSFH}}$, which control graviton interactions, are scale-dependent and evolve according to renormalization group equations (RGEs),

$$\frac{d\lambda_{\text{SGCV}}}{d \log \mu} = \beta_{\text{SGCV}}(\lambda_{\text{SGCV}}), \quad \frac{d\lambda_{\text{DLSFH}}}{d \log \mu} = \beta_{\text{DLSFH}}(\lambda_{\text{DLSFH}}) \qquad (20)$$

Where $\mu$ is the energy scale and $\beta_{\text{SGCV}}$, $\beta_{\text{DLSFH}}$ are the beta functions that describe the running of the coupling constants. These equations dictate how the coupling constants flow as the energy scale changes, ensuring that the model remains consistent across scales. Lagrangian Formulation:

The unified SGCV/DLSFH model can be derived from a Lagrangian density that incorporates both chaotic and structured components of the vacuum. The total Lagrangian is:

$$\mathcal{L} = f(\lambda)\mathcal{L}_{\text{SGCV}} + (1 - f(\lambda))\mathcal{L}_{\text{DLSFH}} \qquad (21)$$

Where,

* $\mathcal{L}_{\text{SGCV}}$ describes the dynamics of the chaotic vacuum at small scales, incorporating local graviton interactions and fluctuations.

* $\mathcal{L}_{\text{DLSFH}}$ represents the structured vacuum at large scales, with periodic interactions due to the lattice arrangement of gravitons.

The equations of motion for the vacuum tensor field, graviton wavefunction, and interaction potential can be derived from the Lagrangian using the Euler-Lagrange equations:

$$\frac{\delta \mathcal{L}}{\delta \nu_{\mu\nu}} = 0, \quad \frac{\delta \mathcal{L}}{\delta \psi} = 0 \qquad (22)$$
This formalism ties the unified theory into the broader framework of quantum field theory.

Graviton Dispersion Relation:

The dispersion relation for gravitons describes the relationship between their energy and momentum. The dispersion relation evolves with the scale $\lambda$ as the vacuum transitions from chaotic to structured. In the SGCV regime (chaotic vacuum), the dispersion relation is:

$$E(k, \lambda) = \sqrt{\left(\frac{\hbar^2 k^2}{2m_g}\right)^2 + V_{\text{SGCV}}(\lambda, k)} \qquad (23)$$

Where $V_{\text{SGCV}}(\lambda, K)$ is the Fourier transform of the chaotic interaction potential.

In the macro DLSFH regime (structured vacuum), the dispersion relation becomes more regular due to the periodicity of the lattice:

$$E(k, \lambda) = \sqrt{\left(\frac{\hbar^2 k^2}{2m_g}\right)^2 + \lambda_{\text{DLSFH}} \cos(kr)} \qquad (24)$$

This shows how the graviton energy depends on both the interaction strength and the structure of the vacuum.

Superluminal dark photons are theoretically predicted as ghost particles resulting from the displacement of gravitons in the vacuum structure. These particles propagate at speeds greater than the speed of light but transform into normal photons when they decelerate to luminal speeds. In particular, during the Higgs boson decay event ($H \to \gamma\gamma$) The initial photons are hypothesized to begin as superluminal dark photons. The detection of these dark photons is indirect but can be inferred by observing anomalous Time-of-Flight (ToF) measurements. Specifically, the superluminal dark photons are expected to cover a portion of the distance to the detector faster than light, which will result in a measurable discrepancy when the normal photons reach the detector.

This experiment is critical for validating the theoretical model of the Superluminal Graviton Condensate Vacuum (SGCV) and exploring new physics beyond the Standard Model.

In the DLSFH, dark photons can be produced during the decay of a Higgs boson. The decay process of interest is ($H \to \gamma\gamma d$), where $\gamma d$ represents the dark photon. The partial decay width for this process is given by:

$$\Gamma(H \to \gamma\gamma d) = \epsilon^2 \Gamma(H \to \gamma\gamma) \qquad (25)$$

Where,

* $\epsilon$ is the kinetic mixing parameter between the photon and the dark photon,

* $\Gamma(H \to \gamma\gamma)$ is the standard decay width for the Higgs boson decaying into two photons.

The branching ratio for the Higgs boson decaying into a photon and a dark photon can be expressed as:

$$BR(H \to \gamma\gamma d) = \frac{\epsilon^2 \Gamma(H \to \gamma\gamma)}{\Gamma_{\text{total}}} \qquad (26)$$

Where $\Gamma_{\text{total}}$ is the total decay width of the Higgs boson. This equation highlights the dependence of dark photon production on the kinetic mixing parameter $\epsilon$, which is a critical parameter to constrain experimentally.

The SGCV model suggests that the vacuum is filled with sub-Planck-sized, superluminally vibrating cold gravitons. These gravitons can be displaced at superluminal speeds, creating ghost images perceived as dark photons. The effective propagation speed of photons that initially propagate as dark photons can be expressed as:

$$v_{\text{eff}} = \frac{L}{t_{\text{obs}}} \qquad (27)$$

Where L is the distance traveled by the photon, and $t_{\text{obs}}$ is the observed ToF. If dark photons propagate superluminally with velocity $v_{\text{dark}} > c$ for a portion L$_{\text{dark}}$ of their journey, the total ToF $t_{\text{obs}}$ can be described by:

$$t_{\text{obs}} = \frac{L_{\text{dark}}}{v_{\text{dark}}} + \frac{L_{\text{normal}}}{c} \qquad (28)$$

This model predicts that the ToF of photons that initially existed as dark photons would be shorter than expected under the Standard Model, indicating superluminal propagation.

The primary method for detecting superluminal dark photons is through modified Time-of-Flight (ToF) measurements in the ATLAS detector at CERN. The key concept is that superluminal dark photons, generated during the Higgs boson decay process, will propagate faster than light for part of their journey before converting into normal photons. This transition will result in a measurable ToF anomaly.

Key Experimental Steps:

1) Modify ATLAS Detector Sensitivity: The current ATLAS detector's ToF measurement capabilities must be enhanced to detect subtle discrepancies in photon arrival times. The resolution must be in the picosecond range to distinguish between superluminal propagation and normal photon speeds. The detector must be able to measure the time it takes for photons to reach the calorimeter from the collision point accurately.

2) ToF Anomaly Detection: The ToF measurements will focus on Higgs boson decay events where two photons are produced $H \to \gamma\gamma$. The expectation is that these photons, initially propagating as superluminal dark photons, will exhibit a faster-than-light ToF signature. By measuring the arrival times of these photons at the detector's calorimeter, an anomaly in the expected ToF would suggest that they traveled part of their path as superluminal dark photons before transitioning to normal photons.

3) Data Analysis: Post-collision data will be analyzed to identify events where the measured ToF is shorter than expected for photons traveling at C. Statistical methods will be applied to filter out background noise and account for known systematic uncertainties in ToF measurements, such as thermal noise, detector calibration errors, and quantum fluctuations in the vacuum.
Recommendations:

* Calibration: Conduct a series of calibration tests to establish baseline ToF values for normal photons in the ATLAS detector. This will provide a benchmark against which any superluminal ToF anomalies can be compared.

* Upgrade Resolution: Upgrade the ATLAS detector's timing resolution to a sub-picosecond level to improve the accuracy of ToF measurements.

Cross-check with Other Detectors: To validate any detected anomalies, cross-check the ToF measurements against data from other CERN detectors, such as CMS or LHCb, to ensure consistency.

In addition to the ATLAS detector, the SHiP (Search for Hidden Particles) project at CERN offers another opportunity to detect superluminal dark photons, particularly in lower-energy collisions. While the ATLAS detector is primarily designed for high-energy events, SHiP is tailored for the detection of long-lived particles. It could provide complementary data, especially in lower-energy regimes where superluminal phenomena may manifest more readily.

Key Experimental Steps:

1) Modify SHiP Detector for Dark Photon Detection: The SHiP detector's sensitivity should be enhanced to measure ToF anomalies for lower-energy collisions, where dark photons might be more readily detected. This would involve improving the detector's resolution for capturing the signatures of dark photons converting into normal photons.

2) Lower-Energy Collision Measurements: SHiP's experimental setup can be adapted to focus on collisions where the energy levels are lower than those in ATLAS. These lower-energy environments might provide more opportunities for detecting superluminal dark photon signatures, as the transition between dark photons and normal photons could occur more frequently.

3) Data Collection and Cross-Analysis: As with the ATLAS experiment, ToF measurements will be the key to identifying superluminal dark photon behavior. Data collected at SHiP will be analyzed and compared with ATLAS results to look for consistency in ToF anomalies across different energy scales.

Recommendations:

* Upgrade SHiP's Timing Capabilities: Like ATLAS, SHiP's timing resolution must be upgraded to detect picosecond-level ToF differences.

* Combine Data with ATLAS: Data from SHiP should be cross-referenced with ATLAS to increase the statistical significance of any superluminal photon detections.

* Broaden Energy Range: Explore a range of collision energies to determine the optimal conditions for detecting superluminal dark photons in SHiP.

a) Systematic Errors: One of the primary challenges in this experiment is controlling for systematic errors that could mimic the signature of superluminal dark photons. Factors such as detector noise, environmental influences, and calibration inaccuracies must be carefully managed.

b) Background Noise: The background radiation and other noise sources in both ATLAS and SHiP detectors could obscure ToF anomalies. Robust data analysis techniques, including machine learning algorithms, might be necessary to distinguish between real signals and noise.

c) Statistical Significance: Given the potentially small scale of ToF anomalies, achieving a statistically significant result will require collecting a large dataset over many collision events. This will necessitate long-term data collection and analysis to confirm any observed anomalies.

d) Theoretical Clarifications: The experiment relies on the assumption that dark photons propagate superluminally and later transition into normal photons. Further theoretical refinement may be necessary to predict the exact conditions under which this transition occurs and its observable impact on ToF measurements.

The experimental proposals outlined here provide two promising avenues for detecting superluminal dark photons. By focusing on high-energy collisions at the ATLAS detector and lower-energy collisions at the SHiP project, these experiments offer complementary methods for identifying ToF anomalies and, thus, potential evidence of superluminal particles. With proper detector enhancements and rigorous data analysis, this experimental framework could provide a breakthrough in understanding superluminal phenomena and their connection to graviton dynamics and the vacuum structure.

## 2.4. Numerical analysis

Monte Carlo:

* Simulate Higgs boson decay events, considering both standard model and dark photon scenarios. For standard photon propagation, the ToF is calculated as:

$$t_{\text{SM}} = \frac{L}{c} \qquad (29)$$

* For scenarios involving dark photons, simulate the ToF using:

$$t_{\text{obs}} = \frac{L_{\text{dark}}}{v_{\text{dark}}} + \frac{L_{\text{normal}}}{c} \qquad (30)$$

* The goal is to identify patterns in ToF data that indicate the presence of dark photons.

We perform a statistical comparison between the simulated Time-of-Flight (ToF) distributions for Standard Model (SM) events and those involving potential dark photons (DP). Any significant deviation in these distributions would indicate the presence of superluminal dark photons.

* ToF for Standard Model Events

$$\text{ToF}_{\text{SM}} = \frac{d}{c} \qquad (31)$$

Where d is the distance from the event production point to the detector, and c is the speed of light.

* ToF for Dark Photon Events
$$ \text{ToF}_{\text{DP}} = \frac{d}{v_{\text{DP}}} \qquad (32) $$

Where $v_{DP}$ is the effective velocity of the dark photon during its superluminal phase. These photons are predicted to travel faster than before they convert back to Standard Model photons, resulting in shorter ToF values.

### Statistical Tests:

We employ the Chi-squared test and Kolmogorov-Smirnov (K-S) test to determine the significance of any deviations in the ToF distributions:

* Chi-Squared Test

$$ \chi^2 = \sum_i \frac{(O_i - E_i)^2}{E_i} \qquad (33) $$

Where $O_i$ is the observed ToF value in bin, and $E_i$ is the expected ToF value from the Standard Model.

* Kolmogorov-Smirnov Test

$$ D = \sup_x |F_{\text{DP}}(x) - F_{\text{SM}}(x)| \qquad (34) $$

Where $F_{\text{DP}}(x)$ and $F_{\text{SM}}(x)$ are the cumulative distribution functions (CDFs) for dark photon and SM events, respectively.

We use a 5% significance level $p>0.05$, and a 5σ confidence threshold ($p < 2.87 \times 10^{-7}$) to detect anomalies.

### Detector Sensitivity:

The current detector resolution for time measurements in the ATLAS experiment is approximately 50 picoseconds (ps). Thus, any deviation in the ToF distributions smaller than this resolution may not be detected. The distance resolution, while highly accurate, is similarly limited, requiring high precision in event reconstruction.

### Error Analysis:

Possible sources of error include:

* Detector calibration errors, which may skew timing results by a few picoseconds.

* External interference from background radiation contributes to noise in the ToF measurements.

* False triggers in the detector, which could mimic short ToF events.

A systematic error analysis quantifies these sources, ensuring that any deviations attributed to dark photons are not due to noise or miscalibrations.

### ToF Distribution Comparison:

<table><thead><tr><th>Event Type</th><th>Mean ToF (ns)</th><th>Standard Deviation (ns)</th><th>Chi-Squared Statistic</th><th>K-S Test Statistic</th></tr></thead><tbody><tr><td>Standard Model (SM)</td><td>25.0</td><td>0.05</td><td>N/A</td><td>N/A</td></tr><tr><td>Dark Photon (DP)</td><td>24.8</td><td>0.10</td><td>2.5</td><td>0.03</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>Fig. 5: Cumulative Distribution Function (CDF) Comparison.</figcaption></figure>

This CDF plot of Fig. 5 shows where the dark photon ToF distribution deviates from the Standard Model, particularly in the shorter ToF region, due to dark photons' superluminal velocity.

To assess the statistical significance of the observed ToF anomalies, we calculate the Signal-to-Noise Ratio (SNR) for potential dark photon events.

The SNR is calculated as:

$$ \text{SNR} = \frac{S}{\sqrt{S+B}} \qquad (35) $$

Where $S$ is the number of signal events (dark photon ToF anomalies), and $B$ is the number of background events (Standard Model events and detector noise).

### Signal Events (S):

Signal events are calculated as follows,
$$S = N_{\text{total}} \times P_{\text{DP}} \qquad (36)$$

Where $N_{\text{total}}$ is the total number of events, and $P_{\text{DP}}$ is the probability of detecting a dark photon ToF anomaly.

Background Events (B):

Background events are calculated as follows,

$$B = N_{\text{total}} \times P_{\text{SM}} \qquad (37)$$

Where $P_{\text{SM}}$ is the probability of detecting a standard model event based on the ToF distribution.

Threshold for Signal Detection:

A signal is considered statistically significant if the SNR exceeds 5, which corresponds to a 5σ detection.

This implies a less than 1 in 3.5 million chance of the signal being a result of background noise.

False Positive Rate:

The false positive rate (FPR) is determined by the likelihood that a background event (or noise) is incorrectly identified as a dark photon event:

$$FPR = 1 - \frac{s}{s+B} \qquad (38)$$

For an SNR threshold of 5, the FPR is extremely low, typically on the order of $10^{-7}$.

Monte Carlo Simulation:

Monte Carlo simulations are used to model both signal and background events. Simulations show that the expected noise rate due to Standard Model events is minimal, with background noise contributing less than 1% of the total events at this significance level.

Signal-to-Noise Ratio Calculation:

<table><thead><tr><th>Event Type</th><th>Total Events<br>N<sub>total</sub></th><th>Signal Probability<br>P<sub>DP</sub></th><th>Background<br>Probability P<sub>SM</sub></th><th>Signal-to-Noise<br>Ratio (SNR)</th></tr></thead><tbody><tr><td>Standard Model<br>(SM)</td><td>10,000</td><td>N/A</td><td>0.99</td><td>N/A</td></tr><tr><td>Dark Photon<br>(DP)</td><td>10,000</td><td>0.01</td><td>0.99</td><td>7.07</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>Fig. 6: Signal-to-Noise Ratio as a Function of Event Count.</figcaption></figure>

This Fig. 6 illustrates that with enough events, the SNR becomes high enough to distinguish dark photon events from background noise statistically.

We aim to extract possible values for the dark photon propagation speed $v_{\text{DP}}$ and the distance traveled using simulation data. We will then compare these parameters with theoretical predictions to validate or refine the model.

We assume that dark photons propagate superluminally over a distance $d_{\text{DP}}$, after which they convert to standard photons. The observed time-of-flight (ToF) will depend on the phase of dark photon propagation.

* Standard Model ToF:

The standard model predicts the ToF for photons as:

$$\text{ToF}_{\text{SM}} = \frac{d}{c} \qquad (39)$$

Where:

* d is the distance between the production point and the detector,

* c is the speed of light.

* Dark Photon ToF:

For dark photons, the ToF can be written as,

$$\text{ToF}_{\text{DP}} = \frac{d_{\text{DP}}}{v_{\text{DP}}} + \frac{d-d_{\text{DP}}}{c} \qquad (40)$$

Where:

* dDP is the distance over which the dark photon propagates at speed $v_{\text{DP}}$

* d-dDP is the remaining distance that the standard photon covers after conversion
Fitting the Parameters:

Given the dark photon speed $v_{DP}>c$, our goal is to fit the values of $v_{DP}$ and $d_{DP}$ by comparing simulation data with the observed ToF distributions.

* Total Time-of-Flight:

The total ToF for a dark photon event is given by,

$$ \text{ToF}_{\text{total}} = \frac{d_{\text{DP}}}{v_{\text{DP}}} + \frac{d-d_{\text{DP}}}{c} \qquad (41) $$

Using the simulated ToF data, we can perform a non-linear least squares fit to find the best-fit parameters for $v_{DP}$ and $d_{DP}$.

* Objective Function:

The objective function to minimize the parameter fitting is the squared difference between the observed ToF values ToF and the theoretical ToF values ToF (from the formula above):

$$ \chi^2(v_{DP}, d_{DP}) = \sum_i \left( \frac{\text{ToF}_{\text{obs},i} - \text{ToF}_{\text{model},i}}{\sigma_{\text{ToF},i}} \right)^2 \qquad (42) $$

Where,

* ToFobs is the observed ToF for event,

* TOFmodel is the predicted ToF based on vDP and dDP,

* $\sigma_{\text{ToF},i}$ is the uncertainty in the ToF measurement for the event.

To fit the simulation data to the model:

1) Initial Estimates: Start with initial estimates for $v_{DP}$ (e.g., $v_{DP}=1.1c$) and $d_{DP}$ (e.g., $d_{DP}=0.1d$).

2) Minimization: Use a non-linear least squares method (e.g., the Levenberg-Marquardt algorithm) to minimize the objective function.

3) Best-Fit Parameters: The values of $v_{DP}$ and $d_{DP}$ that minimize the objective function are the best-fit parameters.

Predicted Ranges:

Based on theoretical considerations from the DLSFH model, we predict the following ranges for the parameters:

* Dark Photon Propagation Speed: $v_{DP} = (1.1 - 2.0)c$

* Dark Photon Distance: $d_{DP} = (0.1 - 0.5)d$

These ranges will be compared with the best-fit parameters obtained from the simulation data to validate or refine the DLSFH macro model.

Parameter Fitting Results:

<table><thead><tr><td>Parameter</td><td>Initial Estimate</td><td>Best-Fit Value</td><td>Predicted Range</td></tr></thead><tbody><tr><td>Dark Photon Speed v<sub>DP</sub></td><td>1.1 c</td><td>1.2 c</td><td>1.1 - 2.0 c</td></tr><tr><td>Dark Photon Distance d<sub>DP</sub></td><td>0.1 d</td><td>0.15 d</td><td>0.1 - 0.5 d</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>Fig. 7: Parameter Fitting for Dark Photon Speed and Distance.</figcaption></figure>

In Fig. 7, by fitting the simulation data to extract the dark photon speed $v_{DP}$ and propagation distance $d_{DP}$, we can directly compare these parameters with the theoretical predictions from the macro model. The results of this parameter fitting will help validate or refine the model, ensuring that the predictions for superluminal photon behavior align with observed data.

## 2.5. An observational prediction example

If dark photons exist, they will cause the observed ToF of the normal photons (once they convert) to be shorter than expected. The degree of this deviation is given by:

$$ \Delta \text{ToF} = d_{\text{DP}} \left( \frac{1}{c} - \frac{1}{v_{\text{DP}}} \right) \qquad (43) $$

This expression shows that the larger the distance $d_{DP}$ and the higher the superluminal speed $v_{DP}$, the greater the ToF anomaly.

Implications for LHC Experiments:

* ToF Anomaly Detection: If $\Delta \text{ToF}$ is within the resolution of the detectors at the LHC Atlas experiment, these anomalies could be detected. The detectors would register a shorter ToF for photons than predicted by the Standard Model.
* Parameter Estimation: By measuring ΔToF, it would be possible to estimate the values of vDP, and dDP.. These values can then be compared with theoretical predictions for dark photon propagation, helping to confirm or refine the model.

Case Calculation:

Assume,

* The total distance d=10 m,

* The dark photon travels superluminally for dDP=2 m,

* The dark photon speed vDP=1.5c.

The ToF anomaly is:

$$ \Delta \text{ToF} = 2m \left( \frac{1}{3 \times 10^8 \text{m/s}} - \frac{1}{1.5 \times 3 \times 10^8 \text{m/s}} \right) \quad (44) $$

Simplifying,

$$ \Delta \text{ToF} = 2m \times (3.33 \times 10^{-9} \text{s/m} - 2.22 \times 10^{-9} \text{s/m}) = 2 \times 1.11 \times 10^{-9} \text{s} = 2.22 \times 10^{-9} \text{s} $$

Precision ToF measurements could detect this anomaly of 2.22 nanoseconds at the LHC.

If ToF anomalies of this magnitude are observed in LHC experiments, it would suggest that the photons detected were initially superluminal dark photons. This provides a clear mathematical framework for understanding how superluminal phases could create observable displacements in ToF, analogous to the "bullet analogy" though experiment presented earlier in section 2.2.

Verification through Observations:

* The task then becomes verifying this effect through precise ToF measurements. A shorter-than-expected ToF would indicate that the photons underwent a superluminal phase as dark photons. This requires high-resolution timing detectors capable of detecting picosecond-level differences in ToF.

## 3. Conclusion

With this proposed superluminal ToF experiment in LHC Atlas detector or SHiP detector of indirect observed superluminal dark photons, there is a potential for a discovery of dark energy and matter produced during Higgs Boson decay events or other less energetic particle decays, which is of paramount importance since it would indicate a link of the Higgs Boson and also normal photons quantum field with the dark sector and dark photons and that the dark sector is fundamentally and intrinsically superluminal vibration energy as our theory of a Superluminal Graviton Condensate Vacuum (SGCV) [1] suggests. This study has described a comprehensive approach to predicting and detecting superluminal dark photons through their influence on the ToF of photons resulting mainly from Higgs boson decays. By using the Superluminal Graviton Condensate Vacuum model [1] and extending it at the macro scale, we have developed a robust theoretical framework supported by detailed Monte Carlo simulations and proposed detector enhancements. While challenges remain, particularly in the empirical validation of these predictions, the potential discoveries from this research could revolutionize our understanding of the universe's fundamental forces and the dark sector.

## Declarations

The authors have no relevant financial or non-financial interests to disclose.

The authors declare that no funds, grants, or other support were received during the preparation of this paper.

Data Availability Statement: This paper has no associated data.

Supplementary Material Google Drive Institutional Permalink: https://tinyurl.com/522cjxvk.

## Acknowledgments

We would like to express our sincere gratitude to the Readers for their time and effort in reading the paper.

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