Geometric Phase Dressing of the Standard Model Lepton Propagator

Geometric Phase Dressing of the Standard Model Lepton Propagator: SMEFT Matching and g−2Anomalies Damian Pikor and Paweł Kurzawski (Dated: November 19, 2025) The persistent discrepancy between the Standard Model (SM) prediction and the experimental measurement of the muon anomalous magnetic moment, ∆ aµ = aexp µ−aSM µ≈ 2 . 51 × 10 −9 , suggests the existence of non-trivial physics at the Compton scale. In this work, we propose that the phenomenological Zitterbewegung (ZBW) of the Dirac electron can be effectively resummed into a physical, covariant Phase Form Factor ( Fϕ ), representing a geometric dressing of the SM vacuum. Unlike standard charge form factors, Fϕ leaves the electromagnetic charge distribution point-like ( FEM ≡ 1), consistent with high-energy scattering limits, but encodes vacuum coherence via a world-surface geometry Σ. We demonstrate that this geometric dressing provides a UV-motivated geometric realization of the dimension-6 dipole operators of the Standard Model Effective Field Theory (SMEFT), mapping the geometric phase radius ⟨r2⟩ϕ to the Wilson coefficients Ceγ . Using the observed ∆ aµ as a calibration input, we derive a Universal Phase Radius benchmark which naturally predicts a negligible electron anomaly (∆ ae∼ 10 −14 ), suppressed by the mass-squared ratio ( me/mµ ) 2 , and a substantial, testable tau anomaly (∆ aτ∼ 7 × 10 −7 ), suggesting a qualitative connection to heavy-flavor anomalies ( R ( D∗ )). Furthermore, we show that the model satisfies stringent Lepton Flavour Violation (LFV) bounds from the MEG II experiment through natural flavor alignment ( | ( ⟨r2⟩ϕ ) µe|≪⟨r2⟩ϕ|µ ) and remains safe from Electric Dipole Moment (EDM) constraints under a CP-even dressing assumption. Finally, we identify a unique falsifiable signature: a momentum-dependent loss of coherence in low-energy electron interferometry, measurable via the visibility slope defined by Fϕ(q2). I. INTRODUCTION The Standard Model (SM) of particle physics has withstood decades of experimental scrutiny. However, as precision measurements reach unprecedented sensitivity, significant tensions have emerged, particularly within the lepton sector. The most enduring of these is the anomalous magnetic moment of the muon, aµ = ( g− 2) µ/ 2. The long-standing discrepancy between the SM prediction and the experimental value, confirmed by the Fermilab Muon g− 2experiment to a combined significance of 4 . 2 σ [ 1 , 2 ], strongly suggests the existence of new physics at the electroweak or Compton scales. Furthermore, persistent hints of Lepton Universality Violation (LUV) in heavy-flavor decays—such as the R ( D )and R ( D∗ )anomalies in b→cτν transitions—indicate that the third generation of leptons may couple to the vacuum or new degrees of freedom differently than the lighter generations [ 3 ]. Conventional solutions to these anomalies typically rely on extending the particle content of the SM. Models involving leptoquarks, Z′ bosons, or supersymmetric partners are frequently invoked to generate the necessary loop corrections to the lepton-photon vertex. In this work, we explore an alternative, complementary hypothesis: that the observed anomalies arise not from new heavy particles, but from a non-perturbative geometric structure of the Standard Model lepton dressing itself. We revisit the phenomenon of Zitterbewegung (ZBW), the rapid oscillatory motion of a free fermion predicted by the Dirac equation. Historically, ZBW has often been dismissed as a coordinate artifact of the single-particle interpretation, vanishing in the Foldy-Wouthuysen representation [ 4 ]. However, within Quantum Field Theory (QFT), the bare electron is unphysical; the physical electron is always dressed by a cloud of virtual photons and pairs. We postulate that ZBW is the phenomenological manifestation of this dressing—a physical, resonant phase geometry of the vacuum fields surrounding the lepton worldline. We model this structure as a covariant world-surface Σ, characterized by a coherence density with a natural scale set by the Compton wavelength, ¯ λC = ℏ/mc . We emphasize that this framework does not replace QED or QFT; rather, it provides a physical, geometric realization of the effective dipole operators that emerge in the low-energy limit of the Standard Model. Crucially, we formalize this geometric framework within the rigorous language of Effective Field Theory (EFT). We introduce a Phase Form Factor, Fϕ ( q2 ), which encodes the coherence properties of the lepton’s internal geometry. To satisfy stringent constraints from high-energy scattering experiments (LEP, LHC), we enforce a strict sector separation: the electromagnetic charge form factor remains point-like ( FEM ≡ 1), while the phase form factor carries the non-trivial geometric information ( Fϕ = 1). In this paper, we demonstrate that this geometric ansatz is mathematically equivalent to a specific pattern of dimension-6 dipole operators in the Standard Model Effective Field Theory (SMEFT) [ 5 ]. SMEFT provides the standard model-independent language to interpret precision data [ 6 , 7 ]. We derive an exact dictionary mapping the geometric phase radius ⟨r2⟩ϕ to the Wilson coefficients Ceγ in the Warsaw basis. By calibrating the model to the observed muon anomaly ∆ aµ , we identify a Universal Phase Radius benchmark. This scenario naturally resolves the hierarchy of anomalies: it predicts a negligible electron anomaly (∆ ae∼ 10 −14 ) 2 and a substantial tau anomaly (∆ aτ∼ 10 −7 ). Finally, we discuss the phenomenological consistency of the model. We show that the geometric dressing naturally respects Lepton Flavour Violation (LFV) bounds, such as µ→eγ , through an intrinsic flavor alignment mechanism. Unlike many BSM theories testable only at high-energy colliders, our model offers a unique falsifiability condition at low energies via precision electron interferometry. II. THEORETICAL FRAMEWORK In this section, we formalize the hypothesis that the lepton is dressed by a coherent phase geometry. We construct a covariant framework that separates charge and phase observables and derive the low-energy observables. A. Assumptions and Sector Separation To reconcile the non-local nature of Zitterbewegung with the point-like behavior of leptons observed in high-energy scattering, we introduce a strict sector separation ansatz. We decompose the physical lepton state into two distinct sectors with independent form factors: 1. Charge Sector ( FEM ): The electromagnetic charge distribution is described by a density ρEM (x). Consistent with QED precision tests and LEP/LHC scattering data, we assume the charge radius is negligible compared to the Compton scale. Formally: ρEM(x)=δ(3)(x) =⇒FEM(q2)≡1.(1) This ensures that the particle couples to the photon as a point charge at tree level. 2. Phase Sector ( Fϕ ): The vacuum coherence surrounding the particle is described by a scalar density ρϕ ( r ), representing the time-averaged probability distribution of the Zitterbewegung resonance. This sector possesses a non-trivial geometric structure characterized by the Compton scale ¯ λC . It generates a Phase Form Factor Fϕ ( q2 ), which modifies the coherence properties of the propagator loop corrections. B. Covariant Resonance World-Surface (Σ) We postulate that the physical electron is associated with a 2-dimensional world-surface Σembedded in Minkowski spacetime, carrying the phase information. The dynamics of Σare governed by a reparametrization-invariant action functional SΣ, inspired by Helfrich-Willmore models [8]: SΣ[X, φ] = ZΣ d2ξ√−hT+κ 2(2H−C0)2+ ¯κKG +hφ 2hab∂aφ∂bφ+λres(2Hℓ −1)2.(2) Here, H is the mean curvature, KG is the Gaussian curvature, and C0 is the spontaneous curvature. The resonance constraint anchors the geometry to the particle’s mass scale, ℓ≡¯ λC = ℏ/mc . Minimization of this action leads to stable, axially symmetric profiles for the coherence density ρϕ(r). C. The Phase Form Factor and Shape-Blindness The Phase Form Factor is defined as the three-dimensional Fourier transform of the normalized radial coherence density ρϕ(r). In the rest frame: Fϕ(q2)=4πZ∞ 0 dr r2ρϕ(r)sin(qr) qr ,(3) where q = p|q2| is the momentum transfer and the density is normalized such that 4 πRr2ρϕ ( r ) dr = 1. Expanding the spherical Bessel function j0 ( qr )for small q , we obtain the infrared (IR) expansion: Fϕ(q2)=4πZ∞ 0 dr r2ρϕ(r)1−(qr)2 6+... = 1 −q2 6⟨r2⟩ϕ+O(q4).(4) Here, ⟨r2⟩ϕ = 4 πRdr r4ρϕ ( r )is the mean-squared phase radius. This expansion demonstrates IR Shape-Blindness: at low momentum transfer, the detailed functional form of ρϕ ( r )is irrelevant. The physics is governed solely by the Phase Radius, ⟨r2⟩ϕ. D. Relation to the Anomalous Magnetic Moment The phase dressing acts as a vertex correction in the non-relativistic limit. By coupling the extended phase geometry to the electromagnetic field, the finite size of the phase cloud induces a Pauli interaction term. The magnitude of the anomalous magnetic moment aℓ is directly proportional to the phase variance. We derive the master formula: ∆a(ϕ) ℓ=m2 ℓ 3⟨r2⟩ϕ.(5) This relation implies that for a fixed geometric scale, the anomaly scales quadratically with the lepton mass m2 ℓ. 3 012345 0 0.5 1 IR Region (g−2) Momentum Transfer q2[arb. units] Phase Form Factor Fϕ(q2) IR Shape-Blindness Geometric Ansatz Spectral Ansatz Figure 1. Illustration of IR Shape-Blindness. Two distinct density profiles (Geometric vs. Spectral) with the same phase radius ⟨r2⟩ϕ yield indistinguishable form factors in the infrared regime (shaded region), diverging only in the UV. Since g− 2 is a zero-momentum observable, it is insensitive to the specific UV completion. III. MATCHING TO SMEFT AND LEFT To rigorously interface our framework with modern phenomenology, we map the geometric parameters onto the Standard Model Effective Field Theory (SMEFT) [5,6]. A. SMEFT Dipole Operators In the Warsaw basis of SMEFT [ 5 ], contributions to lepton magnetic dipole moments arise from dimension6 operators involving the lepton doublet Lp , the righthanded singlet er, and the Higgs doublet H: Opr eγ = (¯ Lpσµν er)HFµν .(6) After Electroweak Symmetry Breaking (EWSB), where ⟨H⟩ = (0 , v/√2 ) T with v≈ 246 GeV, this generates an effective Lagrangian term in the Low-Energy EFT (LEFT): Ldipole eff =Ceγ,ℓℓ v √2Λ2¯ ψℓσµν PRψℓFµν +h.c. (7) B. The Geometric-SMEFT Dictionary The contribution of this effective operator to the anomalous magnetic moment is given by the standard QFT relation [11]: ∆aSMEFT ℓ=2√2mℓv eΛ2ℜ[Ceγ,ℓℓ].(8) This normalization follows the conventions where lepton dipole operators in SMEFT are matched onto LEFT at the electroweak scale mW . We note that between the high scale Λand the electroweak scale, the Wilson coefficients evolve according to the full SMEFT Renormalization Group Equations (RGEs), mixing with other operators [ 12 , 13 ]. We equate the prediction from the phase-geometric dressing (Eq. 5) with the SMEFT prediction (Eq. 8). The matching condition requires ∆ a(ϕ) ℓ = ∆ aSMEFT ℓ . Solving for the Wilson coefficient yields an exact Geometric-SMEFT Dictionary: ℜ[Ceγ,ℓℓ(µ)] = e mℓΛ2 6√2v⟨r2⟩ϕ.(9) This equation provides a bidirectional mapping: experimental data for ∆ aℓ fixes the required size of the vacuum dressing ⟨r2⟩ϕ , which in turn determines the necessary Wilson coefficient. IV. NUMERICAL ANALYSIS AND RESULTS We performed a numerical analysis using two distinct ansätze for the density ρϕ ( r ): a geometric Gabriel’s Horn profile and a spectral Bessel profile. Both were calibrated to the experimental muon anomaly. A. Calibration on the Muon We adopt the experimental value ∆ aexp µ≈ 2 . 51 × 10 −9 . Using Eq. (5), we inverse-solve for the physical phase radius of the muon dressing: ⟨r2⟩ϕ|µ=3·∆aexp µ m2 µ≃6.74 ×10−7GeV−2.(10) Converting this to a root-mean-square (RMS) length: rrms,µ =q⟨r2⟩ϕ|µ≈8.21 ×10−4GeV−1.(11) Comparing this to the muon Compton wavelength ( ¯ λC,µ ≈ 9 . 46 GeV−1 ), the ratio is rrms,µ/¯ λC,µ ≈ 0 . 87 × 10 −4 . The vacuum phase resonance is highly compact, explaining why it has evaded detection in standard scattering experiments. B. Predictions for Electron and Tau Assuming the vacuum dressing size is a fundamental constant of the geometry (the Universal Phase Radius benchmark), we have ⟨r2⟩ϕ|e = ⟨r2⟩ϕ|τ≡ ⟨r2⟩ϕ|µ . The anomaly then scales strictly as m2 ℓ . A detailed discussion of this benchmark is provided in Sec. V A. 4 Table I. Summary of Model Predictions (Universal Radius Benchmark). † indicates calibration input. The values for Ceγ are effective coefficients at Λ = 1 TeV derived from the geometric dictionary. Lepton (ℓ) Mass [GeV] ∆aℓPrediction Ceγ (Λ = 1 TeV) Electron 5.11 ×10−45.9×10−14 1.1×10−17 Muon 0.106 2.51 ×10−9†9.6×10−11 Tau 1.777 7.10 ×10−75.1×10−5 C. Interferometric Visibility A distinct feature of the model is its testability via matter-wave interferometry. The fringe visibility V ( q ) maps to the modulus of the phase form factor: V ( q ) ≈ |Fϕ ( q2 ) | . Analyzing simulated visibility data (file zmg_cl_ir150_interf.csv ), we performed a linear fit in the IR regime: V(q)≈1+sfit q2,with sfit ≈ −0.2496.(12) Using the relation ⟨r2⟩dimless = − 6 sfit , we recover ⟨r2⟩ ≈ 1 . 498, matching the theoretical geometric moment ≈ 1 . 500. This confirms that interferometry can independently measure the g−2generating parameter. 05·10−20.1 0.15 0.2 0.96 0.98 1 Momentum Transfer Squared q2[arb. units] Visibility V(q)≈ |Fϕ| Interferometric Visibility Slope (IR Regime) Theory Fit (s≈ −0.25) Simulated Data Figure 2. Simulated interferometric visibility V ( q )as a function of momentum transfer squared. The linear slope in the IR regime directly recovers the geometric phase radius ⟨r2⟩ϕ≈ 1 . 5 (dimensionless), demonstrating the falsifiability of the model via electron interferometry. V. PHENOMENOLOGICAL CONSTRAINTS We now examine the model’s compatibility with other precision observables, ensuring that the geometric dressing hypothesis remains robust against current experimental bounds. A. Universal Phase Radius Benchmark A particularly simple and instructive limit of our framework is the universal phase radius scenario, where the phase-space dressing is identical for all lepton flavours, ⟨r2⟩ϕ|e = ⟨r2⟩ϕ|µ = ⟨r2⟩ϕ|τ≡ ⟨r2⟩univ ϕ . In this case, the phase-geometric contribution to the anomalous magnetic moment scales purely as m2 ℓ . Taking the measured muon anomaly ∆ aµ = 2 . 51 × 10 −9 as input, we obtain ⟨r2⟩univ ϕ≃ 6 . 74 × 10 −7 GeV −2 . For the electron and tau this immediately implies: ∆auniv e=m2 e 3⟨r2⟩univ ϕ≈5.9×10−14,(13) ∆auniv τ=m2 τ 3⟨r2⟩univ ϕ≈7.1×10−7.(14) The prediction ∆ auniv e is two orders of magnitude below present experimental sensitivities, fully compatible with the effectively null result for ae . Meanwhile, ∆ auniv τ lies comfortably inside our benchmark interval and well below current experimental constraints. This naturalness of the hierarchy ∆ ae≪ ∆ aµ≪ ∆ aτ emerges automatically from the mass hierarchy and the universal dressing geometry. 10−410−310−210−1100 10−14 10−12 10−10 10−8 10−6 e(pred) µ(input) τ(pred) Exp. Bound Lepton Mass mℓ[GeV] Anomaly ∆aℓ Universal Phase Radius Benchmark Model Prediction (∝m2 ℓ) Figure 3. The “Universal Phase Radius” benchmark. The solid points represent the model predictions assuming a flavorindependent vacuum dressing size. The line shows the m2 ℓ scaling. Calibrating on the muon (red circle) naturally predicts a negligible electron anomaly (gray square) and a large tau anomaly (green triangle), consistent with current bounds. B. LFV Bounds and Alignment In the charged-lepton mass basis, the most general electromagnetic dipole interaction in LEFT takes the form 5 [14]: LLEFT ⊃¯ ℓiσµν (Aij +iBijγ5)ℓjFµν +h.c. (15) where i, j = e, µ, τ . The diagonal coefficients Aii control the magnetic moment anomalies, while off-diagonal coefficients Aij induce Lepton Flavour Violation (LFV). In our phase-geometric picture, it is natural to extend the diagonal identification to the full flavour matrix: Aij(mℓ)∼ −emi 12 (⟨r2⟩ϕ)ij.(16) Using the current MEG II limit BR ( µ→eγ ) < 4 . 2 × 10 −13 [9], we obtain the constraint: |(⟨r2⟩ϕ)µe|≲9.1×10−11 GeV−2.(17) Comparing with the diagonal muon phase radius, we find: |(⟨r2⟩ϕ)µe| ⟨r2⟩ϕ|µ ≲1.4×10−4.(18) This quantitatively confirms that the phase-radius matrix must be highly flavor-aligned, in perfect agreement with our benchmark fits where we take (⟨r2⟩ϕ)ij diagonal. C. CP Phases and Electric Dipole Moments The same SMEFT dipole operator Oeγ can also induce CP violation if its Wilson coefficient is complex. In SMEFT we write Ceγ,ℓℓ = CRe ℓℓ + iCIm ℓℓ . The EDM is proportional to the imaginary part: dℓ∼e 2mℓℑv √2Λ2Ceγ,ℓℓ.(19) To estimate the potential impact, consider the muon with Λ=1TeV and maximal CP violation ( |ℑC| ∼ |ℜC| ). The induced muon EDM is approximately dµ≈ 4 . 7 × 10 −28 e·cm . This is many orders of magnitude below the current experimental limit |dµ|< 1 . 8 × 10 −19 e·cm [ 15 ]. Thus, we adopt a CP-even dressing assumption ( ℑC = 0) which is phenomenologically safe. D. Connection to Heavy-Flavor Anomalies The observed tensions in R ( D )and R ( D∗ )suggest a potential violation of lepton universality in b→cτν transitions. Within SMEFT, these are often fitted using scalar or tensor operators. Under RG evolution, tensor operators mix into dipole operators. In our geometric framework, we predict a large tau dipole anomaly ∆ aτ∼ 7 × 10 −7 , corresponding to a large Wilson coefficient Ceγ,ττ . While pure dipole operators do not directly dominate the b→cτν amplitude, they enter SMEFT global fits through operator mixing and correlated constraints [ 10 ], so a large ∆ aτ is naturally correlated with enhanced tau-specific interactions in the heavy-flavor sector. A full global SMEFT fit including R(D(∗))constraints is left for future work. VI. DISCUSSION AND CONCLUSION We have proposed a resolution to the lepton anomalous magnetic moment puzzle by reinterpreting Zitterbewegung as a physical, resonant geometric dressing of the Standard Model vacuum. By introducing a covariant Phase Form Factor Fϕ , we have constructed a model that is mathematically equivalent to the dipole sector of SMEFT in the infrared. Our analysis yields three key conclusions: 1. Consistency: The model reproduces ∆ aµ exactly by calibration, while naturally suppressing ∆ ae and LFV rates due to mass-scaling and flavor alignment. 2. Prediction: We predict a tau anomaly ∆ aτ∼ 7 × 10 −7 , offering a potential link to heavy-flavor anomalies. 3. Falsifiability: Unlike typical BSM theories, our model predicts a momentum-dependent loss of coherence in electron wavepackets, testable via the visibility slope in precision electron interferometry. We conclude that the “missing” form factor in lepton physics may not be a new particle, but a signature of the geometric coherence of the vacuum itself. ACKNOWLEDGMENTS Numerical datasets supporting the SMEFT mapping and interferometric simulations are available in the supplementary repository. Appendix A: Proof of Covariance We construct the phase density ρϕ ( r )from Poincaré scalars. Defining a covariant density D ( x )on the worldsurface Σ, the radial density is obtained by projecting D ( x )onto the hypersurface orthogonal to the 4-velocity uµ using the projector Pµν = ηµν −uµuν . Since all ingredients are covariant, the resulting form factor Fϕ ( q2 ) depends only on the invariant q2. Appendix B: Ansätze Derivations We utilize two profile classes: 1. Geometric (Gabriel’s Horn): Derived from minimizing curvature energy, regularized by UV/IR cutoffs. 6 2. Spectral (Bessel): Models the coherence as a resonant mode in momentum space. Both satisfy the normalization 4 πRr2ρϕdr = 1 and exhibit shape-blindness in the IR limit. [1] B. Abi et al. (Muon g-2 Collaboration), Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm, Phys. Rev. Lett. 126, 141801 (2021). [2] T. Aoyama et al.,The anomalous magnetic moment of the muon in the Standard Model, Phys. Rep. 887, 1–166 (2020). [3] G. Martinelli, S. Simula, and L. Vittorio, Semileptonic B→D∗ decays from light to τ leptons: the extraction of the form factor F2 from data, Eur. Phys. J. C 85, 242 (2025). [4] D. Hestenes, The Zitterbewegung Interpretation of Quantum Mechanics, Found. Phys. 20, 1213 (1990). [5] B. Grzadkowski, M. Iskrzyński, M. Misiak, and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 10, 085 (2010). [6] I. Brivio and M. Trott, How to use the Standard Model effective field theory, Phys. Rept. 793, 1–98 (2019). [7] J. Aebischer et al.,The Standard Model effective field theory at work, Prog. Part. Nucl. Phys. 135, 104074 (2023). [8] W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments, Z. Naturforsch. C 28, 693 (1973). [9] MEG II Collaboration, Search for µ+→e+γ with the First Dataset of the MEG II Experiment, Phys. Rev. Lett. 132, 021801 (2024). [10] A. Crivellin et al.,LHC tau-pair production constraints on aτand dτ, SciPost Phys. 16, 048 (2023). [11] A. Crivellin and M. Hoferichter, Consequences of chirally enhanced explanations of ( g− 2) µ for h→µµ and Z→µµ , JHEP 07, 135 (2021). [12] E. E. Jenkins, A. V. Manohar, and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators I: Formalism and lambda Dependence, JHEP 10, 087 (2013). [13] J. Helset, M. Paraskevas, and M. Trott, Automated oneloop computations in the SMEFT, Phys. Rev. D 103, 096024 (2021). [14] M. Raidal et al.,Flavour physics of leptons and dipole moments, Eur. Phys. J. C 57, 13–182 (2008). [15] G. W. Bennett et al. (Muon g-2 Collaboration), Improved Limit on the Muon Electric Dipole Moment, Phys. Rev. D80, 052008 (2009). [16] W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, Neutrino Trident Production: A Powerful Probe of New Physics with Neutrino Beams, Phys. Rev. Lett. 113, 091801 (2014).