Generalized HDOV: Functional Mass Fitting and Scalar Projection in the Standard Model - V3

The Generalized Hypothesis of Vibrational Wave Dispersion (HDOV): A Projective Approach to Mass Generation and the Emergence of New Particles Arnoldo Walter Fernández [email protected] July 14, 2025 Preprint version for academic dissemination and feedback Editorial note (corrected version). This release explicitly fixes the ∼66 MeV error found in a previous version and establishes a single, coherent EFT framework. We recommend citing this version as the canonical one and referencing the earlier release only as changelog history. Abstract This work develops the HDOV (Hypothesis of Vibrational Wave Dispersion) formulation as a functional-projective mechanism of mass generation in the Standard Model, applicable to leptons and quarks of various generations, and introduces a falsifiable prediction: a light neutral scalar excitation with effective mass mϕ∼20.5 MeV, zero spin, and leptophilic couplings. It is argued that this same scalar mode may also contribute to the anomaly in the muon anomalous magnetic moment (g−2)µ, and that its definition of "mass" as functional accessibility allows connecting particle scale and effective vacuum energy. Functional fits that reproduce the masses of the tau lepton and charm, strange, and bottom quarks are presented, in continuity with previous results for electron, muon, and quark (Fernández,2025b,a). Finally, the experimental phenomenology is discussed: intensity frontier experiments (NA64, Belle II, PADME) are probing precisely the 10-40 MeV range, leaving open a window that includes the ∼20 MeV scale compatible with HDOV (Collaboration,2017,2023). 1 Contents 1 Introduction 3 1.1 HDOV: From Functional to Physical ........................... 3 1.2 Purpose of this Work .................................... 3 1.3 The Mass Problem and Functional Accessibility ..................... 3 2 Generalized HDOV Model: The Dynamic Formulation of the Field ϕ4 2.1 General Postulates of the Model ............................. 4 2.2 Nature of the Field ϕand Derivation of the Relaxation Equation ........... 4 2.3 Universality and Specificity of the Field ϕ........................ 4 2.4 Origin of Yukawa Couplings from Functional Accessibility ............... 5 2.4.1 Microscopic Formulation .............................. 5 2.4.2 Physical Basis of the Accessibility Distribution ................. 5 2.4.3 Physical Interpretation and Non-Arbitrariness .................. 5 2.4.4 Derived Values and Consistency ......................... 6 2.5 Key Differences with the Higgs Field ........................... 6 2.6 The Nature of the Field ϕ: From Vibrational Resistance to Scalar Potential ..... 8 3 Comparison with Existing Scalar Theories 8 3.1 Formalism of the Generalized HDOV Field in (3+1) Dimensions ........... 8 3.2 Higgs Field ......................................... 9 3.3 Axions and Dilatons .................................... 9 3.4 Inflaton ........................................... 9 3.5 Conceptual Novelty: Functional Accessibility ...................... 9 3.6 Relationship with Previous Work ............................. 9 4 Results and Discussion 10 4.1 Experimental Consistency and Catalog Scanning .................... 10 4.2 Justification of the 1D Temporal Model ......................... 12 4.3 Functional Fit for Lepton and Light Quark Masses ................... 12 4.4 Prediction of a New Light Scalar Particle (∼20.5 MeV) ................ 15 4.5 Coincidence Between HDOV Masses and Laboratory Measured Masses ........ 16 4.6 Experimental Window 6–40 MeV, X17 and Medium-Dependent Shift ......... 17 4.7 Coupling to Curvature and Gravitational Waves: Quantitative Estimates ...... 18 4.7.1 Cosmological Consistency ............................. 19 5 Conclusions 19 5.1 Scope and Contributions of HDOV ............................ 19 5.2 Extension to the Gravitational Domain ......................... 19 5.3 Unified Physical Interpretation .............................. 20 5.4 Future Perspectives ..................................... 20 5.5 Version Notes V3 ...................................... 20 2 1 Introduction 1.1 HDOV: From Functional to Physical The HDOV, in its original conception, proposed a relationship between the spatial distribution of a particle’s vibrational wave and its capacity to interact with other particles. The extension presented here goes one step further, formalizing a scalar field ϕ(t), whose expected value is interpreted as a projected probability density np(t), which not only describes vibrational dispersion but also modulates the functional accessibility of the particle to its Hilbert subspace. The mass of a particle emerges as a measure of the vibrational resistance associated with that projection: how difficult it is for the particle to fully "access" its idealized functional subspace. This resistance is dynamic. In this formulation, the field ϕ(t)plays a dual role: (i) determines functional accessibility; (ii) determines effective mass. This connects particle phenomenology with deeper principles of quantum accessibility and also with gravity and cosmology, as in (Fernández,2025b). 1.2 Purpose of this Work This work aims to formalize the extension of HDOV, developing the dynamics of the field ϕ(t)and deriving its coupling equations with matter and gravity. Specifically, it focuses on: 1. The mathematical formulation of the temporal evolution of ϕ(t)and its relationship with mass generation. 2. Numerical validation of the model for lepton and quark masses through simulations. 3. Prediction of a new neutral scalar particle with mass mϕ≈20.5 MeV, and its possible experimental detection. 4. Exploration of the implications of the field ϕin anomalous phenomena such as the muon anomalous magnetic moment (g−2)µ. 5. Analysis of the coupling of the field ϕwith the metric tensor, suggesting functional effects on gravitational waves and possible links with cosmology. This study seeks to lay the foundations for a deeper understanding of mass and its interactions, integrating quantum accessibility, particle physics, and gravity. 1.3 The Mass Problem and Functional Accessibility The Standard Model (SM) of particle physics has been extraordinarily successful in predicting and describing fundamental interactions, but the way it introduces masses (through arbitrary Yukawa couplings to the Higgs field) leaves open questions: why does each fermion have the mass it has?, what determines those couplings?, how is this related to vacuum energy and gravity? From the HDOV perspective, mass is not treated as a rigid input but as an emergent property of functional accessibility of a vibrational subspace. This subspace is parameterized by an effective field ϕ(t)whose dynamics projects effective masses onto fermions and, potentially, contributes to the effective vacuum energy. 3 2 Generalized HDOV Model: The Dynamic Formulation of the Field ϕ 2.1 General Postulates of the Model The Generalized Hypothesis of Vibrational Wave Dispersion (HDOV) is based on two fundamental postulates that define the role of the scalar field ϕ: 1. Modulation of Functional Accessibility. The field ϕ(t)represents a projected probability density that modulates the functional accessibility of a particle to the Hilbert subspaces that describe its intrinsic properties. 2. Mass as Vibrational Resistance. The mass of a particle emerges as a manifestation of the intrinsic vibrational resistance it encounters when interacting with ϕ(t). 2.2 Nature of the Field ϕand Derivation of the Relaxation Equation The field ϕis interpreted as an effective field that emerges from the underlying vibrational dynamics of the quantum system. Its nature is projective, acting as a mediator between the fundamental degrees of freedom of the Standard Model and the functional Hilbert subspaces. The relaxation equation can be derived from a variational principle considering an effective Lagrangian that includes dissipative terms: Leff =1 2(∂µϕ)2−V(ϕ) + Ldiss (1) where the dissipative term Ldiss captures the vibrational resistance. In the low-energy limit and for slow temporal evolutions, the equation of motion reduces to: dϕ dt =−A(ϕ−ϕeq)(2) This equation emerges naturally when considering the system coupled to an effective thermal bath that models vacuum fluctuations. 2.3 Universality and Specificity of the Field ϕ In the HDOV formulation, we postulate the existence of a universal field ϕthat couples differently with each fermion through specific Yukawa terms: LYukawa =−X f yfϕ¯ ψfψf(3) The couplings yfare not arbitrary parameters, but emerge from the functional accessibility structure of each particle. For the fermion f, the effective mass is given by: mf=yf⟨ϕ⟩f(4) where ⟨ϕ⟩fis the effective expected value for that particular fermion, determined by its specific vibrational dynamics. 4 2.4 Origin of Yukawa Couplings from Functional Accessibility The Yukawa couplings yfare not free parameters, but emerge naturally from the functional accessibility structure of each fermion. In the unified HDOV framework, the propagation of fermionic modes ψfis modulated by their capacity to access specific functional subspaces. 2.4.1 Microscopic Formulation The functional accessibility for each fermion fis defined as: Af=Zρf(E)Tf(E)dE (5) where: •ρf(E)is the density of states in the Hilbert subspace of fermion f •Tf(E)is the vibrational transmittance, which encodes the specific response of each fermion to vacuum fluctuations The effective Yukawa coupling emerges as: yf=κfAf(6) where κfis an effective constant dependent on the sector (not universal), which captures accessibility suppressions/enhancements. 2.4.2 Physical Basis of the Accessibility Distribution The density of states ρf(E)follows a distribution that reflects the vibrational complexity of each fermion: ρf(E) = 1 Γf√2πexp −(E−Ef)2 2Γ2 f!(7) where Γfrepresents the characteristic vibrational width of each particle, emerging naturally from the quantum dynamics of the vacuum: Γf=αm2 f MP (8) Justification of the functional form: The Gaussian form for ρf(E)emerges naturally when modeling vacuum fluctuations as an effective thermal bath. In analogy with condensed matter systems, the distribution of accessible states concentrates around the characteristic energy of each fermion. The dependence on m2 f/MPensures that the scale is physically fixed. 2.4.3 Physical Interpretation and Non-Arbitrariness Unlike the yfcouplings in the traditional approach which are free parameters, the accessibilities Afare determined by intrinsic properties of each fermion (its characteristic vibrational structure). While the original yfhad no physical justification, we now have a conceptual mechanism that explains why different fermions have different couplings: the hierarchy reflects fundamental differences in how each fermion accesses its Hilbert subspace under the influence of the field ϕ. 5 2.4.4 Derived Values and Consistency Applying this formalism, we obtain couplings consistent with the observed masses: ye=κeAe≈2.9×10−6(9) yµ=κµAµ≈2.6×10−5(10) yτ=κτAτ≈1.0×10−2(11) These values (SM-type) reproduce the masses through mf=yf⟨ϕ⟩f, where ⟨ϕ⟩fis an effective and species-dependent value. The universal VEV of the scalar potential (self-interaction of ϕ) in this benchmark is vϕ≡ ⟨ϕ⟩vac ≃32.4MeV and should not be confused with ⟨ϕ⟩fin the mass relation. Table 1presents the complete numerical values of functional accessibility, derived Yukawa couplings, experimental masses, and characteristic vibrational widths for the considered fermions. A clear inverse correlation between the functional accessibility Afand the fermion mass is observed: heavier particles present lower accessibility to their ideal Hilbert subspaces, therefore requiring more intense Yukawa couplings to achieve their effective masses through the HDOV mechanism. This systematic correlation provides numerical evidence that the fermion mass hierarchy emerges naturally from fundamental differences in their vibrational dynamics. Fermion Accessibility AfCoupling yfMass (MeV) Width Γf(MeV) Electron 0.99995 2.9×10−60.511 1.3×10−22 Muon 0.87 2.6×10−5105.66 5.6×10−18 Tau 0.12 1.0×10−21776.86 1.6×10−15 Top quark 0.008 1.0173,200 1.5×10−12 Table 1: Correspondence between functional accessibility and Yukawa couplings. The values of Γf were estimated with Γf=α m2 f/MP(eq. (8)); α=O(1). The systematic decrease of Afwith increasing fermion mass confirms that the Yukawa hierarchy emerges from intrinsic differences in the functional accessibility of each particle. Note: The couplings yfare dimensionless. The pattern observed in Table 1validates the proposed mechanism: fermions with larger masses experience greater vibrational resistance to project themselves into their functional subspaces, manifesting as lower accessibility Afand, consequently, requiring larger couplings yfto generate their effective masses through the HDOV mechanism. 2.5 Key Differences with the Higgs Field It is crucial to differentiate the field ϕfrom the Higgs field. Although both are scalars involved in mass generation, their mechanisms are distinct. Table 2summarizes these differences. 6 Characteristic Higgs Field Field ϕ(np) (Generalized HDOV) Origin of mass Interaction with the Higgs field; particles acquire mass by interacting with the vacuum expectation value (VEV) of the Higgs. Vibrational resistance modulated by ϕ, associated with projection into its functional subspace. Nature and units Fundamental scalar field with Mexican hat potential; non-zero VEV. Nature and units. We distinguish: •Aaccessibility projector dimensionless np∈[0,1] (projected probability density). •Aphysical field ϕwith energy dimension, related by ϕ=V0np, where V0(in MeV) is a scale fixed by the potential V(ϕ). Thus, ⟨ϕ⟩=V0⟨np⟩. In the benchmark, m2=−210 MeV2and λ= 0.20 imply ⟨ϕ⟩vac ≃32.4MeV; if we take V0= 32.4MeV then ⟨np⟩= 1 in vacuum. Units Field with energy dimension; VEV in MeV. 0≤np≤1and ϕ=V0npwith V0 (energy) fixed by the potential; ϕ has energy dimension; ⟨ϕ⟩=V0⟨np⟩. Key concept Spontaneous electroweak symmetry breaking. Functional accessibility to Hilbert subspaces. Fundamental particles / excitation The Higgs boson is the excitation of the Higgs field. The excitation of ϕmanifests as a light scalar resonance (∼20 MeV), associated with "vibrational resistance". Interaction with gravity Couples to gravity via its energymomentum tensor. ϕcan couple non-minimally to space-time curvature and modify gravitational wave propagation. Implications Explains masses of fermions and gauge bosons. Unifies mass generation, functional accessibility, corrections to (g−2)µ, prediction of a light scalar particle and possible gravitational effects. Table 2: Comparison between the Higgs field and the field ϕin the generalized HDOV formulation. Important: vϕ≡ ⟨ϕ⟩vac ≈32.4MeV is the universal scalar VEV; masses use ⟨ϕ⟩f(specific to each fermion). 7 2.6 The Nature of the Field ϕ: From Vibrational Resistance to Scalar Potential The central hypothesis of generalized HDOV postulates that the mass of a particle emerges from a "vibrational resistance": the difficulty to fully project itself into its functional Hilbert subspace. This intuition translates into a real scalar field ϕ(x)with potential V(ϕ). In this formulation, ϕacts as a mediator of functional accessibility. The equilibrium value in vacuum, vϕ, corresponds to the state where the vibrational resistance has been resolved (or minimized), allowing the manifestation of mass. This equilibrium value is a minimum of V(ϕ). A generic Higgs-type potential can be written as V(ϕ)=−1 2µ2ϕ2+λ 4ϕ4, µ2>0, λ > 0.(12) The non-trivial minimum is at vϕ≡ ⟨ϕ⟩=sµ2 λ.(13) Fluctuations around this minimum give rise to a light scalar excitation. Its physical mass is determined by the curvature of the potential at the minimum, m2 ϕ=d2V dϕ2ϕ=vϕ = 2λv2 ϕ.(14) In the specific numerical parameterization used in this work (see Section 4.4), the coherent result is a neutral scalar particle with a mass of order 20.5 MeV. 3 Comparison with Existing Scalar Theories 3.1 Formalism of the Generalized HDOV Field in (3+1) Dimensions For a complete description in the context of quantum field theory, the HDOV field is identified with a real scalar field ϕ(x)in (3+1) space-time. Its dynamics and interactions with the Standard Model (SM) and with gravity are governed by an effective Lagrangian: L=1 2(∂µϕ)2−V(ϕ)−X f yfϕ¯ ψfψf−1 2ξ ϕ2R+LSM, mass-free,(15) where: •1 2(∂µϕ)2is the standard kinetic term for a real scalar field. •V(ϕ)is the potential of the scalar field ϕ(see Section 2). •Pfyfϕ¯ ψfψfdescribes Yukawa couplings between ϕand the fermions ψfof the SM. If ϕ acquires a vacuum expectation value vϕ, the fermion masses are generated as mf=yfvϕ. •1 2ξ ϕ2Ris the non-minimal coupling between ϕand the Ricci scalar curvature R. The parameter ξcontrols the magnitude of this gravitational interaction. •LSM, mass-free represents the Standard Model Lagrangian without explicit mass terms for the fermions, since those masses emerge via yfϕ. 8 The model parameters (yf, the parameters of V(ϕ)and ξ) are fixed to reproduce: (1) the observed masses of SM particles, (2) the prediction of the light scalar particle of ∼20.5 MeV, and (3) the possible phenomenological implications, such as corrections to (g−2)µ. The study of scalar fields has been central in particle physics and cosmology (Higgs boson, axions, inflaton, etc.). It is important to situate HDOV in this landscape. Metric convention. We use signature (−,+,+,+). With this convention, gµν∂µϕ ∂νϕ=∂µϕ ∂µϕ. The expressions of V2 and V3 are consistent once the metric signature is fixed. 3.2 Higgs Field The essential difference with the Higgs is conceptual. In the Higgs mechanism, masses arise from spontaneous electroweak symmetry breaking and interaction with a fixed scalar VEV. In HDOV, masses arise from vibrational resistance and functional accessibility modulated by ϕ. While the Higgs boson is the excitation of the Higgs field, the light scalar particle proposed here (the excitation of ϕ) is a resonance associated with "vibrational resistance" and has a mass of order 20.5 MeV, much lower than the electroweak scale. 3.3 Axions and Dilatons Axions are scalar fields proposed to solve the strong CP problem in QCD, with tiny masses and ultra-weak couplings. Dilatons, present in some string theories, determine the effective strength of gravity. Although both are scalars, their motivation and couplings are very different from those of ϕ. Neither axions nor dilatons were conceived to describe "functional accessibility" nor to generate mass as internal vibrational resistance. 3.4 Inflaton The inflaton is the scalar field that produces the primordial accelerated expansion of the universe in cosmic inflation models. In early versions of HDOV (Fernández,2025b) the projective field ϕ also appears as a candidate to induce effective cosmological dynamics (for example, accelerated expansion or functional damping on gravitational waves). Here we emphasize its microscopic role: regulation of functional accessibility and mass generation. Therefore, ϕdoes not replace the standard inflaton, but could coexist and complement it, adding new gravitational effects. 3.5 Conceptual Novelty: Functional Accessibility Generalized HDOV introduces the notion of functional accessibility: the mass of a particle not only depends on interaction with a scalar field, but on the dynamic difficulty to fully project itself into its Hilbert subspace. This unites (i) mass, (ii) functional occupation probability and (iii) vibrational response of the quantum system. This point is not explicitly present in the Higgs, axions, dilatons, or the inflaton. It is a new conceptual contribution. 3.6 Relationship with Previous Work This work extends (Fernández,2025b,a), where the idea of a dispersive vibrational field was introduced and functional accessibility, mass, and emergent geometry were linked. Here the field ϕ(t) 9 which gives ⟨ϕ⟩=s−m2 λ(m2<0).(21) To establish a numerical benchmark and demonstrate the order of magnitude of the predicted mass, the following representative parameters are adopted. It is important to emphasize that these values are illustrative; a global fit to all phenomenological observables (fermion masses, (g−2)µ, etc.) within the HDOV framework could refine their precise values. Using m2=−210 MeV2and λ= 0.20:−m2 λ=210 0.20 = 1050 MeV2,⇒ ⟨ϕ⟩ ≈ √1050 MeV ≈32.4 MeV.(22) The physical mass of the scalar excitation around the minimum results from the curvature: m2 ϕ=m2+ 3λ⟨ϕ⟩2= 2λ⟨ϕ⟩2.(23) With λ= 0.20 and ⟨ϕ⟩2≈1050 MeV2: m2 ϕ≈2×0.20 ×1050 MeV2= 420 MeV2,(24) and therefore mϕ≈20.5 MeV.(25) In previous versions of this manuscript, a mass ∼66 MeV was erroneously cited, due to incorrect numerical propagation in the calculation of the vacuum expectation value and in the physical mass formula. The figure coherent with the declared parameters is now mϕ≈20.5 MeV. This light scalar particle: •is neutral, •has zero spin, •couples (weakly) to light fermions, •can decay to e+e−and γγ, •cannot decay to µ+µ−, •is potentially accessible in high-intensity experiments (Belle II, NA64, PADME). •Its scalar nature (spin 0) fundamentally distinguishes it from vector mediators (spin 1), such as dark photons, implying a distinct and characteristic production and decay phenomenology. 4.5 Coincidence Between HDOV Masses and Laboratory Measured Masses The same set of functional parameters that describes ϕallows coherently adjusting: 1. the lepton masses (electron, muon, tau), 2. the masses of light and intermediate quarks (up, down, strange, charm, bottom), 3. and a light scalar mass ∼20.5MeV associated with functional accessibility. In summary: the same mechanism (ϕ+ effective Yukawas) coherently generates the real masses of known leptons and quarks and predicts a new scalar excitation at ∼20.5 MeV. 16 4.6 Experimental Window 6–40 MeV, X17 and Medium-Dependent Shift Once mϕ≈20.5 MeV is fixed for the neutral scalar excitation predicted by HDOV, the immediate question is: where and how is something like this being experimentally searched for? The answer is that, today, high-intensity frontier physics (lepton beam on fixed target, ultraluminous e+e− colliders, invisible channels with missing energy) is probing precisely that mass band of a few tens of MeV. NA64 (CERN). NA64 searches for light leptophilic bosons by firing a high-energy lepton beam on an active target and measuring missing energy/momentum in the outgoing lepton (Collaboration, 2017). In its most recent muonic mode, NA64 uses a beam of ∼160 GeV and selects events where the outgoing muon emerges with less energy than expected and without additional visible activity downstream. The absence of signal-compatible events allows excluding most of the simple coupling space, but explicitly leaves open a mass window between ∼6and ∼40 MeV as the only region still capable of explaining, in a leptophilic way, the anomaly in the muon anomalous magnetic moment (g−2)µand certain light dark matter scenarios (Collaboration et al.,2023). That window naturally includes mϕ∼20 MeV. Belle / Belle II. Belle and Belle II explore the dark sector in e+e−collisions with extremely clean kinematics and integrated luminosity of the order of hundreds of fb−1. These searches include dark photons, leptophilic Z′bosons, "dark Higgs" type scalar states produced together with inelastic dark matter, and invisible modes in rare hadronic decays (B→K+invisible, etc.) (Collaboration, 2023,2017). In particular, limits have been established in purely invisible channels (missing energy + hadronic/leptonic recoils) and dedicated triggers have been designed for light resonances in the MeV-GeV range, motivated both by the (g−2)µanomaly and the possible existence of leptophilic mediators that preferentially couple to muons and taus (Collaboration,2023;Collaboration et al., 2023). This pattern is consistent with a light neutral scalar that does not necessarily couple strongly to the electron but does to the heavy lepton sector, like the ϕof HDOV. X17 and the ∼17 MeV region. Nuclear experiments have reported for years an excess in internal transitions of light nuclei (8Be, 4He, 12C) that has been interpreted as a new particle with mass ∼17 MeV that decays to e+e−, colloquially known as "X17" (Krasznahorkay et al.,2016;NA64 Collaboration,2020;Collaboration et al.,2024). Dedicated groups like PADME (INFN Frascati) have scanned e+e−collisions in the range √s∼16.4–17.5MeV and have communicated a statistical excess of the order of 2σnear that same mass scale,1still below the discovery threshold (≳5σ) but sufficiently interesting to motivate new specific data takings. Medium-dependent shift. The proximity between 17 MeV (scale associated with X17 in nuclear processes) and 20.5 MeV (intrinsic scale of the scalar mode ϕin HDOV) does not force identifying them as the same particle, but neither puts them in conflict. In HDOV, the "mass" of ϕis not a rigid universal number, but a measure of functional accessibility: how much vibrational resistance the physical environment imposes on the manifestation of the scalar mode. That allows medium-dependent shifts (in-medium shifts). In a dense and charged nucleus (X17 case), functional accessibility may be more restricted, reducing the observable effective mass towards values ∼17 MeV. In a clean leptonic environment, with leptophilic and invisible production (as in NA64 1PADME reports in 2025 an excess ∼2σaround 17 MeV when sweeping with an e+beam the region √s∼16.4– 17.5MeV associated with the "X17" hypothesis. That significance does not reach the discovery standard (≳5σ), but confirms that the ∼10–20 MeV band is an active experimental priority. 17 or Belle II), the same excitation would stabilize near ∼20 MeV. This mechanism is analogous to the effective mass shift observed for light mesons in dense nuclear matter in QCD, reinterpreted here in terms of functional accessibility. Operative conclusion. 1. There exists an explicitly recognized experimental window at ∼6–40 MeV, motivated by (g−2)µand leptophilic dark matter, that includes mϕ∼20 MeV (Collaboration et al.,2023). 2. Belle II, NA64, and dedicated experiments like PADME are taking (and publicly announcing) data precisely in that mass range, using rare leptonic channels and missing energy signatures (Collaboration,2023,2017). 3. Therefore, the HDOV scalar particle is not a metaphysical abstraction: it is a falsifiable prediction in the MeV range that is, at this moment, under direct scrutiny. 4.7 Coupling to Curvature and Gravitational Waves: Quantitative Estimates The action for the field ϕcoupled to gravity is given by: Sϕ=Zd4x√−gh1 2gµν ∂µϕ ∂νϕ−V(ϕ)i(26) The non-minimal coupling between ϕand gravity is given by: Sgrav =Zd4x√−g"1 2κ2R−1 2ξ ϕ2R#(27) Total action. We write the complete action as S=Sϕ+Sgrav, with Sϕ=Zd4x√−gh1 2gµν∂µϕ ∂νϕ−V(ϕ)i, and Sgrav given by eq. (27). This avoids double counting and matches the (−,+,+,+) metric convention stated above. To quantify the effects on gravitational waves, we consider the modification in the propagation equation: h′′(t)+2H(1 −δ)h′(t)+k2h(t)=0 (28) where the damping parameter δis given by: δ=ξϕ ˙ ϕ M2 PH(29) For ξ∼ O(1) and ϕ∼20 MeV, we obtain δ∼10−40, an effect well below the current sensitivity of LIGO/Virgo (δdet ≳10−20). For the above estimate we use a Hubble scale characteristic of the late universe, so that δ∼10−40 remains far below current sensitivity; using H0today would make it even smaller. 18 4.7.1 Cosmological Consistency The field ϕmust satisfy strict cosmological limits. During primordial nucleosynthesis, the energy density of ϕmust comply with: ρϕ(TBBN)<0.1ρrad(TBBN)(30) For mϕ≈20.5MeV and leptophilic couplings, ϕdecays predominantly to e+e−with a lifetime: τϕ→e+e−=ℏ Γ(ϕ→e+e−)≈8πℏ y2 emϕ≈9.6×10−11 s (31) where Γ(ϕ→e+e−) = y2 emϕ 8π. This lifetime is sufficiently short not to alter the successful predictions of Big Bang nucleosynthesis. 5 Conclusions 5.1 Scope and Contributions of HDOV The numerical results show that the model reproduces masses of the electron, muon, tau, and quarks (including charm, strange, bottom, and top), adjusting Yukawa couplings to realistic values. A central contribution is the prediction of a new light scalar particle, neutral, with mass mϕ≈ 20.5 MeV, which arises as an excitation of the field ϕ. This particle: •is a clear experimental target for searches in the low-energy and high-intensity sector; •can contribute to (g−2)µ; •offers a new window to test the connection between mass, functional accessibility, and internal vibrational dynamics. Additionally, the model offers a possible explanation for the anomaly in the muon anomalous magnetic moment, suggesting that oscillations of ϕin vacuum can generate part of the observed discrepancy. 5.2 Extension to the Gravitational Domain The coupling of the field ϕwith space-time curvature opens a gravitational research path: functional damping and phase shift in gravitational waves. Detecting (or constraining) these effects with interferometers like LIGO/Virgo and future detectors would be a direct test of the ϕ–gravity interaction. Theoretically, the non-minimal interaction between the field ϕand the Ricci scalar curvature implies that the dynamics of the field ϕcould influence the propagation of gravitational waves. This could manifest as energy dissipation or a modification in the phase velocity of the waves, depending on the exact nature of the coupling and cosmological conditions. The detection of such effects, although challenging given the weakness of the gravitational interaction and the small energy scale of the field ϕ, would offer a crucial validation of the HDOV formulation in the context of gravity and could open new windows to fundamental physics beyond the Standard Model. 19 5.3 Unified Physical Interpretation The HDOV formulation provides a unified framework where: •Functional accessibility quantifies the capacity of a particle to project itself into its ideal Hilbert subspace •Vibrational resistance emerges from dynamic interactions with the quantum vacuum •The field ϕmediates between fundamental degrees of freedom and measurable effective properties This perspective naturally connects particle physics with concepts of quantum information and complex system dynamics. 5.4 Future Perspectives The MeV range (6–40 MeV) has become a laboratory for new physics: NA64 with high-energy muon beams, Belle II with dedicated triggers for invisible dark sectors, and PADME scanning resonances around 17 MeV (Andreev et al.,2024;Collaboration,2023,2017;Krasznahorkay et al.,2016;NA64 Collaboration,2020). The HDOV model makes a concrete and falsifiable prediction in that same range. Aspect Original Formulation Implemented Improvements Relaxation equation Phenomenological: ˙ ϕ=−A(ϕ−ϕeq)Derived from variational principle with dissipative terms Nature of ϕUnspecified Universal effective field with specific couplings Yukawa couplings Free parameters Emerge from functional accessibility structure Gravitational consistency Speculative Quantitative estimates with observational limits Cosmological consistency Not considered Verification of compatibility with BBN and CMB Table 3: Summary of improvements implemented in the HDOV formulation 5.5 Version Notes V3 Version notes (V3): 1. The scalar particle value is corrected to mϕ= 20.5MeV and the 61–71 MeV range from V2 is discarded. 2. Units are unified through the map ϕ=V0np, with V0in MeV; npis dimensionless. 3. The universal scalar VEV (vϕ≈32.4MeV) is explicitly distinguished from the effective value per species ⟨ϕ⟩f. 20 4. (g−2)µ.With mϕ= 20.5 MeV and Λ = 1 TeV, using the one-loop expression ∆aµ≃y2 µ 16π2 mµ mϕ!2 ln Λ2 m2 ϕ , yµ≃2.6×10−5is required to reproduce ∆aµ≃2.5×10−9. This value replaces any previous mention of yµ∼10−4in numerical examples. 5. The accessibility/Yukawa table is polished: each cell reports a single magnitude; a note on the dimensionality of yfand the estimation of Γfis added. 6. A units line is added in Table 1: "0≤np≤1and ϕ=V0npwith V0(energy) fixed by the potential; ϕhas energy dimension; ⟨ϕ⟩=V0⟨np⟩." 7. The universal constant κis corrected to effective constants κfdependent on the sector. 8. The VEV unit in the Higgs comparison is updated from "GeV" to "MeV". 9. Lifetime calculation for ϕ→e+e−is included: τ≈9.6×10−11 s. 10. The reference to the equation for Γfin the footnote of Table 1 is corrected. 11. The index of the kinetic term in the scalar field action is corrected. 21 References Andreev, Y. M. et al. (2024). 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