The Generalized Hypothesis of Vibrational Wave Dispersion (HDOV): A Projective Framework for Mass Generation and Emergent Geometry - V2

The Generalized Hypothesis of Vibrational Wave Dispersion (HDOV): A Projective Approach to Mass Generation and the Emergence of New Particles Arnoldo Walter Fern´andez [email protected] July 14, 2025 July 14, 2025 (revision v2: November 7, 2025) Editorial Note (corrected version). This version explicitly corrects the ∼66 MeV error that appeared in a previous version and establishes a coherent and unique EFT framework. It is recommended to cite this version as the canonical one and refer to the previous one only as a changelog history. Abstract In this corrected version, we present the EFT formulation of the Generalized Hypothesis of Vibrational Wave Dispersion (HDOV), which unifies functional accessibility in Hilbert subspaces with effective mass generation. The scalar sector is described by a potential with a VEV v≈32.4 MeV and a neutral scalar of mϕ≈20.5 MeV (correcting a previous value of ∼66 MeV). Fermionic mass is modeled by mf=yfvwith effective Yukawa couplings; we discuss its phenomenological consistency without violating perturbativity at low energies. Electromagnetic interactions are parameterized by the effective coefficient Cγ≡cγ/Λ2, allowing confrontation with current production and limits (NA48/2, NA64, Belle II) in the 10–30 MeV window. The impact on (g−2)µwithin a controlled scheme is also considered. This framework consolidates HDOV as a falsifiable and self-consistent hypothesis, and opens extensions towards gravity and cosmology through non-minimal couplings. 1 Contents 1 Introduction 4 1.1 The purpose of this work ................................. 4 1.2 The problem of mass and functional accessibility .................... 4 1.3 HDOV: From functional to physical ........................... 4 2 The Generalized HDOV Model: EFT Formulation 5 2.1 Lagrangian, Potential, and Vacuum Minimum ...................... 5 2.2 Mass Generation and the Emerging Scalar Particle ................... 5 2.3 Electromagnetic Interactions ............................... 6 2.4 Mechanism of Effective Yukawa Couplings ........................ 6 2.5 Key differences with the Higgs field ............................ 8 2.6 HDOV-3 Benchmark and Parameter Scheme ....................... 8 2.7 Stability and Unitarity ................................... 8 2.7.1 Stability for large couplings ............................ 9 3 Detailed Phenomenology and Experimental Predictions 9 3.1 Decay widths of the scalar particle ............................ 9 3.1.1 Decay to fermions: ϕ→f¯ f............................ 9 3.1.2 Decay to two photons: ϕ→γγ .......................... 9 3.1.3 Total width and lifetimes ............................. 10 3.2 Production cross-section in e+e−............................. 10 3.3 Experimental limits and comparison ........................... 11 3.3.1 Comparison with NA48/2 ............................. 11 3.3.2 Sensitivity at NA64 ................................ 11 4 Comparison with Scalar Theories and Alternative Models 12 4.1 Higgs Field ......................................... 12 4.2 Axions and Dilatons .................................... 13 4.3 Inflaton ........................................... 13 4.4 Technicolor and Composite Models ............................ 13 4.5 Conceptual Novelty: Functional Accessibility ...................... 13 4.6 Relation to Previous Works ................................ 13 5 Numerical results and phenomenology 14 5.1 Prediction of the Light Scalar Particle .......................... 15 5.2 Contribution to the Muon Anomalous Magnetic Moment (g−2)extrmmu ...... 15 5.3 Interpretation ........................................ 16 6 Discussion and Conclusions 17 6.1 Discussion: Scope and contributions of HDOV ..................... 17 6.2 Extension to the Gravitational and Cosmological Domain ............... 17 6.2.1 Action and Gravitational Field Equations .................... 17 6.2.2 Cosmological Implications ............................. 17 6.3 Functional Effects on Gravitational Waves ........................ 17 6.4 Concrete testable predictions ............................... 18 6.5 Experimental perspectives and MeV mass window ................... 18 2 6.6 Conceptual differences with other theories ........................ 20 6.7 Implications and next steps ................................ 20 6.8 Limitations and Future Directions ............................ 20 7 Conclusions 20 7.1 Scope and contributions of HDOV ............................ 20 7.2 Extension to the gravitational domain .......................... 21 7.3 Future perspectives ..................................... 21 8 Appendix A: Effective Field Theory and Operator Dimensions 21 8.1 Canonical dimensions in 4D ................................ 22 8.2 Examples in HDOV .................................... 22 8.3 Yukawa rule and self-interactions ............................. 22 3 1 Introduction 1.1 The purpose of this work The present work aims to formalize the extension of HDOV, developing the dynamics of the field ϕ(t) and deriving its coupling equations with matter and with gravity. Specifically, it focuses on: 1. The mathematical formulation of the temporal evolution of ϕ(t) and its relation to mass generation. 2. The numerical validation of the model for lepton and quark masses through simulations. 3. The prediction of a new neutral scalar particle, with mass mϕ≈20.5extrmMeV , and its possible experimental detection. 4. The exploration of the implications of the ϕfield in anomalous phenomena such as the anomalous magnetic moment of the muon (g−2)µ. 5. The analysis of the coupling of the ϕfield with the metric tensor, suggesting functional effects on gravitational waves and possible links with cosmology. This study seeks to lay the groundwork for a deeper understanding of mass and its interactions, integrating quantum accessibility, particle physics, and gravity. 1.2 The problem of mass and functional accessibility The nature of mass in the Standard Model is explained by the Higgs mechanism: the Higgs field acquires a vacuum expectation value (VEV) and, through Yukawa couplings, imparts mass to fermions and vector bosons (Higgs,1964;Weinberg,1967). However, this mechanism does not explain why each mass takes the particular observed value, nor does it link mass to a geometric principle or functional accessibility of the state space itself; nor does it directly address the role of vacuum energy or the global stability of the Higgs potential at cosmological scales. In this work, we propose a functional-projective reformulation: mass is not a fixed input but the result of how much ¨vibrational accessibility¨a dynamic subspace of the ϕ(t) field has when projected onto the observed spacetime. In other words, mass emerges as an effective property of accessibility, not as an arbitrary number imposed by hand. This notion of functional accessibility unifies two traditionally separate planes: •the generation of elementary particle mass, •and the effective contribution to the vacuum energy density. 1.3 HDOV: From functional to physical HDOV, in its original conception, proposed a relationship between the spatial distribution of a particle’s vibrational wave and its ability to interact with other particles. The extension presented here goes a step further, formalizing a field ϕ(t) (projected probability density), which not only describes vibrational dispersion but also modulates the functional accessibility of the particle to its Hilbert subspace. A particle’s mass emerges as a measure of the vibrational resistance associated with that projection: how difficult it is for the particle to ¨ fully access¨ıts idealized functional subspace. This resistance is dynamic. 4 In this formulation, the ϕ(t) field plays a dual role: (i) it determines functional accessibility; (ii) it determines the effective mass. This connects particle phenomenology with deeper principles of quantum accessibility and also with gravity and cosmology, as explored in (Fern´andez,2025b). 2 The Generalized HDOV Model: EFT Formulation We adopt an Effective Field Theory (EFT) framework to describe low-energy physics. The model is built upon a real scalar field, ϕ, with a potential that allows for spontaneous symmetry breaking. 2.1 Lagrangian, Potential, and Vacuum Minimum The minimal effective Lagrangian describing the dynamics of the ϕfield and its interactions is: L=1 2(∂µϕ)(∂µϕ)−V(ϕ)−X f yfϕ¯ ψfψf+LSM, free (1) where the self-interaction potential is chosen in the standard Higgs form: V(ϕ)=−1 2µ2ϕ2+λ 4ϕ4,(µ2>0,λ>0) (2) Note on notation: In later sections, for clarity and consistency with the literature, the notation m2=−µ2will often be used for the quadratic term of the potential, where m2<0 to ensure symmetry breaking. The potential minimization condition, V′(ϕ) = 0, leads to a non-zero vacuum expectation value (VEV): ⟨ϕ⟩=v=sµ2 λ(3) This VEV, v, represents the equilibrium state of the field and is the fundamental scale that determines mass generation in the model. The values v≈32.4 MeV and λ≈0.2 emerge from a phenomenological fit that optimizes the reproducibility of fermionic masses while maintaining consistency with experimental limits. Specifically: •vis determined by requiring that the electron’s Yukawa coupling, ye=me/v ≈0.016, remains perturbative (ye≪1) •λis fixed by the relation mϕ=√2λv to obtain mϕ≈20.5 MeV, a value consistent with unexcluded experimental windows •This set v, λ minimizes the χ2by simultaneously reproducing the masses of e,µ,τand light quarks •The resulting scale v≈32.4 MeV is low enough to avoid conflicts with electroweak precision but sufficient to generate realistic fermionic masses 2.2 Mass Generation and the Emerging Scalar Particle The mass of Standard Model fermions is generated through Yukawa-type couplings with the ϕfield. After symmetry breaking, each fermion acquires a mass: mf=yfv(4) 5 where yfis the Yukawa coupling coefficient for fermion f. Quantum fluctuations of the field around its VEV, defined by ϕ(x) = v+s(x), correspond to a physical particle, s(x). The mass of this scalar particle is determined by the curvature of the potential at the minimum: m2 ϕ=V′′(v) = −µ2+ 3λv2= 2µ2= 2λv2(5) Using the reference VEV value, v≈32.4 MeV, obtained from phenomenological fits, and the consistent parameters of the model, a mass for the scalar particle is predicted: mϕ≈20.5 MeV (6) This prediction is one of the central results of the HDOV model. 2.3 Electromagnetic Interactions Effective parametrization. Electromagnetic interactions are parameterized by the effective coefficient Cγ≡cγ/Λ2for the operator ϕ2FµνFµν. Experimental and precision limits directly constrain Cγ; therefore, we report and compare in terms of Cγ, leaving cγand Λ as UV-dependent parameters. The benchmark Cγ= 10−6extrmMeV −2(for example, cγ= 1, Λ = 1 extrmGeV ) is representative: any pair (cγ,Λ) that reproduces the same Cγhas the same low-energy phenomenology. The interactions of the scalar field ϕwith the electromagnetic sector are described by an effective dimension-six operator, which is the lowest-dimension operator that allows a direct coupling between ϕand photons without violating the gauge symmetries of the Standard Model. This operator is introduced into the Lagrangian as: Lϕγγ =cγ Λ2ϕ2FµνFµν (7) where Fµν is the electromagnetic field strength tensor, cγis a dimensionless coefficient, and Λ is the scale of new physics. This operator is crucial for ϕdecays to photons and for ϕproduction in colliders. It is important to note that, for the base model, the ambiguity of a dimension-five operator has been removed, focusing solely on the dimension-six operator to maintain parsimony and consistency with symmetry breaking. The effective operator cγ Λ2ϕ2FµνFµν introduces the new physics scale Λ. Based on: •Electroweak precision limits: Λ≳1 TeV from LEP/SLD precision measurements •EFT Consistency: Λ>4πv ≈400 MeV for validity of derivative expansion •Low-energy phenomenology: Λ∼110 GeV compatible with collider searches We adopt Λ = 1 GeV as a conservative reference value, which implies cγ/Λ2∼10−6MeV−2 for cγ∼1. This choice avoids conflicts with existing searches while allowing detectable signals in intensity experiments like Belle II and NA64. 2.4 Mechanism of Effective Yukawa Couplings Quantitative UV example. If ∆mf≃cfv v2 EW/Λ2, then yeff f≃∆mf/v ≃cf(v2 EW/Λ2). With vEW = 246 extrmGeV , Λ = 1 extrmTeV and cf∼1, we obtain yeff f≃2462/10002≃0.0605. 6 Effective Yukawas at low energy. Scope of the UV mechanism. With vEW = 246 extrmGeV and Λ = 1 extrmTeV , we obtain yeff f≃cfv2 EW/Λ2≃0.0605 cf. For the top quark (yeff t≃5.3×103), ct∼ O(105) would be required, or Λ ≪vEW, outside the minimal EFT regime. We restrict ourselves to light/medium fermions; the top case demands a specific UV sector (by flavor or higher-dimension operators). In the low-energy regime, we use yeff f=κfmf/v with 0 < κf≤1, where κfparametrizes effective accessibility/projection suppressions; in particular, for the electron channel we consider yeff e= 10−4 (equivalent to κe≃6×10−3for v≃32.4 MeV). The phenomenological relation mf=yf⟨ϕ⟩with ⟨ϕ⟩ ≈ 32.4 MeV successfully reproduces the masses of leptons and quarks. However, a literal interpretation where yfrepresents a fundamental Yukawa coupling would lead to values yf≫1 for heavy fermions (e.g., yt∼5340 for the top quark), outside the perturbative regime and compromising the validity of the EFT expansion. We propose that this apparent breakdown of perturbativity is an artifact of the effective description at low energies. The yfin our EFT Lagrangian should be interpreted as effective couplings (yeff f), which encapsulate the integrated effects of a more fundamental ultraviolet (UV) physics. A natural mechanism, consistent with the EFT philosophy, is that the yeff farise from higherdimension operators involving the Standard Model Higgs field (H). The simplest dimension-6 operator that fulfills this role is: LUV ⊃X f cf Λ2(ϕ†ϕ)¯ ψfH ψf+ h.c.,(8) where Λ is the scale of new physics and cfare perturbative dimensionless coefficients (cf≲1). After spontaneous electroweak symmetry breaking (⟨H⟩=vEW/√2 with vEW ≈246 GeV) and HDOV (⟨ϕ⟩=v), this operator generates a contribution to the fermion mass: ∆mf≈cfv v2 EW Λ2.(9) The effective Yukawa coupling that would be measured at low energies is then: yeff f=∆mf v≈cfv2 EW Λ2.(10) For a scale Λ ∼1 TeV and cf∼1, we obtain yeff f∼10, which allows reproducing the masses of heavy fermions like the charm or bottom quark. For the top quark, whose mass requires yeff t∼5000, the minimal EFT mechanism with Λ ∼1 TeV and cf≲1 is insufficient. ct∼105would be needed for Λ = 1 TeV, violating perturbativity. This suggests that the top quark requires a specific UV sector (non-perturbative or with higher-dimension operators) that goes beyond the scope of this minimal EFT model. Therefore, we restrict ourselves to light and medium fermions where the mechanism is consistent. This mechanism justifies the use of yfvalues as effective parameters in our phenomenological analysis, maintaining the theoretical coherence of the model. The need for a Λ scale in the TeV range for heavy fermions becomes a testable prediction of the HDOV model. 7 2.5 Key differences with the Higgs field Aspect Standard Model (Higgs) Generalized HDOV Origin of mass Gauge symmetry breaking Functional projection and vibration Nature of the field Physical (gauge field) Functional and physical Space of action Internal SU(2) ×U(1) space Hilbert space / spacetime Additional predictions Higgs boson Scalar particle ϕ20.5 MeV Table 1: Comparison between the Higgs model and the generalized HDOV. 2.6 HDOV-3 Benchmark and Parameter Scheme Consistent numerical set. We adopt the set v≃32.4extrmMeV ,λ≃0.20 and m2= −210 extrmMeV 2in the potential V(ϕ) = 1 2m2ϕ2+λ 4ϕ4with m2<0. With this, m2 ϕ= 2λv2and mϕ≃20.5extrmMeV . We define the minimal HDOV-3 benchmark with three free parameters: {µ2, λ, cγ/Λ2}(11) with the effective Lagrangian: LHDOV-3 =1 2(∂µϕ)2−−1 2µ2ϕ2+λ 4ϕ4−X ℓ=e,µ,τ yℓϕ¯ ψℓψℓ−cγ Λ2ϕ2FµνFµν.(12) The derived relations are: v=⟨ϕ⟩=µ/√λ, m2 ϕ= 2µ2= 2λv2, mf=yfv. Table 2: Free parameters of the HDOV-3 benchmark and derived relations Free parameters µ2>0 (with m2=−µ2), λ,cγ/Λ2 Relations ⟨ϕ⟩=pµ2/λ,mϕ=√2µ,yf=mPDG f/⟨ϕ⟩ Production ϕ2F2⇒e+e−→γϕϕ (tree) Dim-5 Extension (c′ γ/Λ)ϕFµνFµν ⇒e+e−→γϕ The dimension-6 operator ϕ2FµνFµν/Λ2constitutes our base choice, while the dimension-5 operator ϕFµνFµν/Λ is considered only as an optional extension to enable single-body production. 2.7 Stability and Unitarity Unitarity bound. A conservative estimate for the perturbative validity of scalar self-interactions is Λunit ∼4π v/√λ≈0.91 extrmGeV for v≃32.4 MeV and λ= 0.20. Potential stability: The potential V(ϕ) = −1 2µ2ϕ2+λ 4ϕ4is stable for λ > 0, with a global minimum at ⟨ϕ⟩=v. The condition of positive vacuum energy is automatically satisfied at the minimum. 8 Unitarity limits: Scalar-scalar scattering amplitudes satisfy partial unitarity up to the scale Λunit ≈0.91 extrmGeV . The ϕϕ →ϕϕ scattering amplitudes at energy Eare bounded by: |M(ϕϕ →ϕϕ)|≲λ 2+9λ2v2 E2−m2 ϕ <8πfor E < Λunit.(13) Radiative stability: One-loop corrections do not destabilize the potential for couplings yf≲ O(1). The main corrections come from fermionic loops: ∆V1-loop ≈ − y4 f 16π2ϕ4log ϕ2 µ2!,(14) which do not develop new minima for yf<4π. Our couplings yf∼mf/v amply satisfy this condition. 2.7.1 Stability for large couplings For large effective Yukawa couplings, as in the case of the top quark (yeff t∼5000), one-loop radiative corrections could destabilize the potential. The approximate stability condition is y4 f/(16π2)≲λ. For λ∼0.2, this requires yf≲4π1/2λ1/4∼5. Since yeff texceeds this limit by orders of magnitude, a UV sector is required to stabilize the potential, possibly through cancellations with contributions from new fermions or scalars in the complete UV theory. 3 Detailed Phenomenology and Experimental Predictions 3.1 Decay widths of the scalar particle The scalar particle ϕcan decay into several channels. We calculate the main decay widths using parameters consistent with HDOV: mϕ= 20.5 MeV, yeff e=κe(me/v) with κe≪1 (where κe parametrizes effective accessibility/projection suppressions), and yµ= 5 ×10−4: 3.1.1 Decay to fermions: ϕ→f¯ f For a fermion fwith mass mfand coupling yf, the decay width is: Γ(ϕ→f¯ f) = y2 fmϕ 8π 1−4m2 f m2 ϕ!3/2 (15) For electrons (yeff e=κeme v(κe≪1), mϕ= 20.5 MeV): Γ(ϕ→e+e−) = (10−4)2×20.5 8π 1−4×(0.511)2 (20.5)2!3/2 = 8.13 ×10−9MeV (16) 3.1.2 Decay to two photons: ϕ→γγ Observation. If the ϕ→γγ channel is to be modeled with a phenomenologically relevant rate, it is convenient to include a dimension-5 operator, ∆L= (c′ γ/Λ) ϕFµνFµν, with effective coupling gϕγγ ≡c′ γ/Λ, which induces Γ(ϕ→γγ)=g2 ϕγγm3 ϕ/(64π). In the absence of this term, the fermionic loop with ye= 10−4gives Γ ∼10−12 MeV, suppressed by α2. 9 The contribution of a CP-even scalar is given by: ∆aµ=y2 µ 8π2Z1 0 dx x2(1 −x)m2 µ x2m2 µ+(1−x)m2 ϕ (31) With mϕ= 20.5 MeV and v= 32.4 MeV, we numerically evaluate the integral, which we call I(mϕ): I(mϕ) = Z1 0 dx x2(1 −x)m2 µ x2m2 µ+(1−x)m2 ϕ≈0.318 (32) The current experimental discrepancy is ∆aexp µ= (2.51±0.59)×10−9(g 2 Collaboration,2021). Equating the theoretical contribution to the experimental one: ∆aµ=κ2 µm2 µ 8π2v2I(mϕ)=∆aexp µ(33) Solving for κµ: κµ=s8π2v2∆aexp µ m2 µI(mϕ)≈2.42 ×10−4(34) This value κµ≈2.42 ×10−4reproduces the central value of the experimental discrepancy ∆aµ. For completeness, we show the behavior of ∆aµas a function of κµ: κµ= 1 ×10−4⇒∆aµ≈0.43 ×10−9 κµ= 2 ×10−4⇒∆aµ≈1.71 ×10−9 κµ= 2.42 ×10−4⇒∆aµ≈2.51 ×10−9 The choice κµ≈2.42 ×10−4exactly reproduces the experimental discrepancy of ∆aexp µ= 2.51 ×10−9. Table 4: Contribution of ϕto (g−2)µfor small values of κµ κµ∆aµ×10−9% of explained discrepancy 1.0×10−40.43 17% 2.0×10−41.71 68% 2.42 ×10−42.51 100% 3.0×10−43.85 153% 5.3 Interpretation The physical reading is direct: mass is not an intrinsic constant, but the reflection of the functional friction that the universe offers to a projected elementary vibration. In this framework, functional accessibility becomes the common origin of: •fermionic masses; •the effective vacuum energy; •the emergence of a neutral scalar mode. 16 6 Discussion and Conclusions 6.1 Discussion: Scope and contributions of HDOV The generalized HDOV unifies: 1. Mass generation through functional projection. 2. Accessibility to Hilbert subspaces. 3. Influence on the emergent geometry of spacetime. Unlike the Higgs, ϕacts both functionally and physically, connecting mass with the vibrational structure of the state space. 6.2 Extension to the Gravitational and Cosmological Domain The HDOV model naturally extends to the gravitational sector through a non-minimal coupling between the ϕfield and the curvature of spacetime. 6.2.1 Action and Gravitational Field Equations The total action of the system (gravity, ϕfield, and Standard Model) is written as: S=Zd4x√−g"M2 P 2R−1 2ξ ϕ2R−1 2∂µϕ ∂µϕ−V(ϕ)+LSM#,(35) where MPis the Planck mass, Ris the Ricci scalar, and ξis the non-minimal coupling parameter. The presence of the ξϕ2Rterm modifies Einstein’s equations, resulting in: M2 P−ξϕ2Gµν =T(ϕ) µν +T(SM) µν +ξ∇µ∇νϕ2−gµν□ϕ2.(36) This implies that the effective gravitational constant depends on the local value of the ϕfield, Geff =GN/(1 −ξϕ2/M2 P). 6.2.2 Cosmological Implications In cosmology, the ϕfield can play a dynamic role. Its expectation value can evolve with time, affecting the expansion history of the universe. An accessibility parameter η(t)∝ ⟨ϕ(t)⟩/MPthat smoothly decreases over time can induce a phase of accelerated expansion, similar to quintessence models. Furthermore, variations of ⟨ϕ⟩during primordial nucleosynthesis could modulate quark masses and reaction rates, leaving observable imprints on the abundances of light elements. Cosmological vacuum stability imposes restrictions on the model. The effective Planck mass, M2 eff =M2 P−ξϕ2, must remain positive. This generally requires ξ > 0 to avoid ¨negative gravity¨ınstabilities in high-field epochs, such as the early universe. 6.3 Functional Effects on Gravitational Waves The ϕ-gravity coupling modifies the propagation of gravitational waves. Starting from action (24) with non-minimal coupling ξϕ2R, the equation for tensorial perturbations hµν in a FriedmannLemaˆıtre-Robertson-Walker background is obtained by varying the action: 17 SGW =Zd4x√−g"M2 P−ξϕ2 21 4∇ρhµν∇ρhµν +···#(37) The modified wave equation results: □hµν +ω2 c2 1−ξϕ2 M2 P!hµν +O(∂ϕ) = 0 (38) Comparing with the proposed form (27), we identify: f(ϕ(x)) = −ξϕ2 M2 P +O ∂µϕ∂µϕ M2 Pω2!(39) For a static field ϕ(r) around a massive object, solving the Klein-Gordon equation in Schwarzschild geometry: ϕ(r)≈ϕ∞1−2GM c2rξ/2 ≈ϕ∞exp −ξGM c2r(40) where ϕ∞is the asymptotic value. This produces cumulative effects on propagation: h(r) ;≈h0exp −ω cZℑ(f(ϕ))drexp iω cZℜ(f(ϕ))dr=h0exp (−Aatten) exp (i∆Φ) with attenuation and phase shift coefficients: Aatten ≈ω c ξϕ2 ∞ M2 P D, ∆Φ ≈ω c ξϕ2 ∞ M2 P D(41) where Dis the propagation distance. 6.4 Concrete testable predictions •Scalar particle ϕ: Mass ∼20.5 MeV, detectable in B →K + X (Belle II) or missing energy events in NA64. •g−2 anomalies: ϕoscillations explain muon deviations. •Gravitational waves: Attenuation and phase shift measurable in GW catalogs with future detectors. 6.5 Experimental perspectives and MeV mass window The generalized HDOV framework makes an immediate falsifiable prediction: the existence of a neutral scalar excitation associated with the ϕfield, with a physical mass of the order of mϕ≃ 20.5extrmMeV , leptophilic couplings, and spin zero. This excitation should be producible in intensity frontier experiments: very high luminosity e+e−colliders, searches with lepton beams on fixed targets, and missing energy modes. Today there are experimental efforts dedicated precisely to this mass range: 18 •NA64 (CERN). NA64 uses directed leptonic beams on an active target to search for light bosons that escape invisibly, leaving as a signature an outgoing lepton with missing energy (Collaboration,2017). In muonic mode, NA64 has explored the production of leptophilic mediators with a beam of ∼160 GeV and identifies events where the outgoing muon loses energy anomalously but nothing else appears in the detector. In the first systematic pass of these data, no signal events were observed, which allows ruling out parts of simpler parameter space, but leaves open a mass window approximately between 20 MeV and 40 MeV as still viable for a leptophilic boson connected to (g−2)µ.1 •Belle / Belle II. Belle and Belle II search for dark photons, leptophilic Z′bosons, and even ’dark Higgs’-like scalar states in e+e−collisions with extremely high luminosity, covering masses from a few MeV up to several GeV (Collaboration,2023). These analyses include invisible channels (missing energy) and rare leptonic modes, exactly the type that would produce a light neutral scalar that couples more strongly to muons/taus than to electrons, and that can decay into modes that are difficult to observe. •∼17 MeV region (“X17”). Various nuclear experiments have reported an excess in internal transitions that is interpreted as a possible new particle of ∼17 MeV decaying into e+e−. Dedicated collaborations like PADME have scanned e+e−collisions in the region √s∼16.4– 18 MeV and reported a statistical excess of the order of 2σnear the same scale, 2without yet reaching the standard discovery level (≳5σ). In HDOV logic, the proximity between the predicted scale mϕ≃20.5 MeV and nuclear signals at ∼17 MeV is not necessarily a contradiction. In this framework, the ’mass’ of the scalar excitation is understood as a measure of functional accessibility: how much vibrational resistance the environment offers for the mode to exist. This implies that the effective mass can undergo medium-dependent shifts. In a dense and charged nucleus (case of ’X17’-type anomalies), functional accessibility is strongly restricted, and the effective frequency of the mode can drop to ∼17 MeV; in a clean or nearly empty leptonic environment, the same excitation would tend to stabilize at ∼20 MeV. This ’medium-dependent shift’ is analogous to the mass shifts known in nuclear physics for light mesons in dense matter. The operational conclusion is twofold: 1. There is an active and still unexcluded experimental window between ∼6 and ∼40 MeV where a neutral leptophilic scalar boson connected to (g−2)µnaturally fits, and this window includes the scale ∼20 MeV predicted by HDOV. 2. Intensity programs (NA64, Belle II, dedicated beams like PADME) are precisely exploring this region with missing energy signatures and rare leptonic channels. Therefore, the HDOV prediction is falsifiable in existing experiments. 1The most recent public results from NA64 in muon mode (∼160 GeV beam) indicate that, after not seeing excesses with missing energy compatible with a new leptophilic boson, the only region that can still simultaneously explain the (g−2)µanomaly and certain light dark matter scenarios is the 6– 40 MeV window. This window naturally includes mϕ∼20 MeV. 2PADME (INFN Frascati) scanned with an e+beam in the region √s∼16.4–17.5 MeV, which coincides with the window indicated by nuclear anomalies attributed to ”X17”, and reported a ∼2σexcess around 17 MeV. It does not reach the discovery threshold (≳5σ), but confirms that masses of the order of tens of MeV are being probed with priority. 19 6.6 Conceptual differences with other theories Unlike axions (Peccei and Quinn,1977) or dilatons (Taylor and Veneziano,1988), ϕis not tied to specific symmetries, but to functional accessibility. Its connection with loop quantum gravity could be explored to understand emergent geometry. 6.7 Implications and next steps •Refine yicouplings for precise observables. •Analyze ϕin cosmology (quintessence). •Search for the ϕparticle in Belle II/NA64, focusing on B →K+Xore−+Z→e−+Z+ϕ channels. The generalized HDOV offers a unifying framework for mass, functional accessibility, and emergent geometry, with falsifiable predictions that connect particle physics and gravity. 6.8 Limitations and Future Directions Despite the advances presented, the generalized HDOV model has certain limitations that open avenues for future research: •Functional accessibility and its formalism: The notion of ¨ functional accessibility¨ıs central but requires a more rigorous mathematical formalism that explicitly connects it with quantum field theory and the geometry of Hilbert spaces. •Heavy fermions and mass hierarchy: Although the mechanism of effective Yukawa couplings allows reproducing the masses of light and medium fermions, the case of the top quark (yeff t∼5000) remains a challenge. This suggests the need for a more complex UV sector or specific higher-dimension operators for heavier fermions, which could imply a hierarchy of new physics scales. •Connection with the Standard Model: While the HDOV model coexists with the Standard Model, a deeper integration that explores possible mixing between the ϕfield and the Higgs boson, or the influence of ϕon gauge coupling constants, could enrich the framework. These limitations do not invalidate the model, but rather position it as a starting point for a more exhaustive exploration of mass generation and fundamental interactions. 7 Conclusions This work has presented an extension of the Generalized Hypothesis of Vibrational Wave Dispersion (HDOV), introducing a dynamic scalar field ϕ(t) that directly links the generation of elementary particle mass with its functional accessibility to Hilbert subspaces. 7.1 Scope and contributions of HDOV Numerical results show that the model reproduces electron, muon, tau, and quark masses (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 ϕfield. This particle: 20 •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. Furthermore, the model offers a possible explanation for the anomaly in the anomalous magnetic moment of the muon, suggesting that ϕoscillations in the vacuum can generate part of the observed discrepancy. 7.2 Extension to the gravitational domain The coupling of the ϕfield with spacetime curvature opens a gravitational research avenue: functional damping and phase shift in gravitational waves. Detecting (or bounding) these effects with interferometers like LIGO/Virgo and future detectors would be a direct test of the ϕ–gravity interaction. 7.3 Future perspectives The lines of work that emerge are: •Extend the model to a complete three-dimensional treatment of ϕ, including spatial distribution and non-local couplings. •Study couplings with gauge bosons and other sectors of the Standard Model. •Characterize in detail the light scalar particle (mϕ≈20.5 MeV): decay modes, production in Belle II, NA64, PADME, high-intensity electron beams, etc. •Re-evaluate the contribution to (g−2)µwith the corrected parameters and adjust limits on yµ. •Deepen the cosmological implications: role of ϕin late accelerated expansion, nucleosynthesis, and large-scale structure. •Develop quantitative predictions of functional damping of gravitational waves and compare them with observational data. 8 Appendix A: Effective Field Theory and Operator Dimensions In this work, “dimension D” refers to the mass dimension of the operator in effective field theory (EFT) in 4-dimensional spacetime. The Lagrangian density must have mass dimension 4. Therefore, if an operator has dimension D > 4, its coefficient carries powers of the cutoff scale Λ that compensate: [OD]=D⇒ L ⊃ c ΛD−4OD. 21 8.1 Canonical dimensions in 4D We use the following dimension assignments: [∂µ]=1,[s]=1,[Aµ] = 1,[Fµν]=2, [ψ] = 3 2,[yf]=0,[λ]=0. 8.2 Examples in HDOV Dimension-6 operator (default base): L(6) EM =cγ Λ2s2FµνFµν,[s2F2]=2×1+2×2 = 6.(42) The first photonic vertex involves two scalars (s2γγ), so in leptonic colliders the natural process is e+e−→γss. Optional dimension-5 extension: ∆L(5) EM =c′ γ ΛsFµνFµν,[sF2]=1+2×2=5.(43) Enables the linear vertex sγγ and the e+e−→γs channel. 8.3 Yukawa rule and self-interactions With the shift ϕ=v+s, mf=yfv, L(s) int ⊃ −λvs3−λ 4s4,(44) and m2 ϕ= 2µ2= 2λv2. These relations fix the low-energy phenomenology without appealing to extra dimensions. 22 References Collaboration, B. I. (2023). 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