Geometric Phase Dressing of the Standard Model Lepton Propagator: SMEFT Matching, Unitary Non-Local Regulator, and g − 2 Anomalies

Geometric Phase Dressing of the Standard Model Lepton Propagator: SMEFT Matching, Unitary Non-Local Regulator, and g−2Anomalies Damian Pikor and Paweł Kurzawski (Dated: November 27, 2025) The persistent discrepancy in the muon anomalous magnetic moment, ∆ aµ≈ 2 . 51 × 10 −9 , suggests the existence of non-trivial physics at the Compton scale. In this work, we explore the hypothesis that the observed anomalies arise not from new particles, but from a non-perturbative geometric structure of the lepton dressing itself. We formalize this by introducing two distinct form factors: (i) a point-like Charge Form Factor, FEM ≡ 1, consistent with high-energy scattering, and (ii) a non-trivial Phase Form Factor Fϕ ( q2 ), encoding a covariant geometric dressing of the SM vacuum. We implement this dressing via a Gaussian representative of the Efimov class of entire function regulators, ensuring perturbative unitarity and ghost-free behavior. Crucially, employing a field redefinition argument, we demonstrate that the theory can be mapped to an effective Weyl geometry where the electromagnetic current remains local in the dressed frame but manifests non-local interactions in the laboratory frame. This framework yields a UV-safe realization of dimension-6 SMEFT dipole operators, mapping the geometric phase radius ⟨r2⟩ϕ to Wilson coefficients Ceγ . Calibrating to ∆ aµ , we derive a Universal Phase Radius benchmark yielding ∆ ae∼ 10 −14 and ∆ aτ∼ 7 × 10 −7 , providing a falsifiable prediction for future precision experiments. I. INTRODUCTION The Standard Model (SM) has withstood decades of scrutiny, yet the muon anomalous magnetic moment ( aµ ) remains a stubborn outlier [ 1 – 4 ]. While heavy-flavor anomalies suggest universality violation [ 9 – 11 ], traditional explanations involving new particles (e.g., Z′ , Leptoquarks) face increasing pressure from high-energy collider bounds (LHC, LEP). In this work, we explore the hypothesis that the observed anomalies arise not from new particles, but from a non-perturbative geometric structure of the lepton dressing itself. Motivated by the inherent non-locality of field interactions at the Compton scale and the necessity of regularizing UV divergences in a unitary manner, we consider a covariant "world-surface" (Σ) describing a geometric phase dressing of the lepton. This perspective aligns with recent proposals treating electromagnetism as a purely geometric theory, where charge density and Zitterbewegung arise directly from the metric structure of spacetime [8]. To reconcile this extended structure with the pointlike behavior observed at high energies, we adopt the formalism of Non-Local Quantum Field Theory (NLQFT) using Efimov-class regulators [ 15 – 18 ]. This allows us to construct a theory that is: 1. IR-Rich: Generates the required ∆ aµ via effective dipole operators. 2. UV-Safe: Remains unitary and ghost-free thanks to the entire function regulator. We map this geometric framework directly to the Standard Model Effective Field Theory (SMEFT) [ 12 – 14 ], providing a rigorous dictionary between geometry and Wilson coefficients. II. THEORETICAL FRAMEWORK A. Non-Local Action and Unitarity We construct a non-local effective action that preserves gauge invariance and unitarity. We postulate that the geometric dressing modifies the kinetic term of the lepton via an entire operator E: LNL =−1 4F2 µν +¯ ψE(D2 µ)(i/ D−m)E(D2 µ)ψ, (1) where Dµ = ∂µ + ieAµ is the covariant derivative. To ensure loop convergence and ghost-free behavior, we choose the Gaussian regulator of the Efimov class: E D2 µ Λ2!= exp −D2 µ 2Λ2!.(2) The regulator scale Λis physically tied to a reference Compton wavelength ¯ λC≡ℏ/ ( mref c ). In the phenomenological analysis below we take mref = mµ and treat ¯ λC as a single geometric length scale characterizing the vacuum phase dressing, common to all charged leptons. To recover the specific form factor width required by the Gaussian vertex in Eq. (B3), we identify Λ = √2 ¯ λC .(3) B. Geometric Phase Sector Before analyzing the field theoretic consequences, it is useful to formulate the phase sector at the level of a covariant coherence density. We distinguish between the electromagnetic charge distribution, which we take to be strictly point-like: ρEM(x)=δ(3)(x) =⇒FEM(q2)≡1,(4) 2 and a non-trivial phase coherence density ρϕ ( r )associated with the geometric dressing. Assuming spatial isotropy in the rest frame, the Phase Form Factor is defined as: Fϕ(q) = 4πZ∞ 0 dr r2ρϕ(r)sin(qr) qr , q ≡ |q|.(5) The low-momentum expansion defines the geometric phase radius: Fϕ(q2→0) = 1 −1 6⟨r2⟩ϕq2+. . . (6) where ⟨r2⟩ϕ≡ − 6 dFϕ dq2|0 . To leading order in q2 , all IR observables are "shape-blind" and depend only on ⟨r2⟩ϕ . C. Sector Separation via Field Redefinition A central structural property of our framework is the exact separation between the geometric dressing (Phase Sector) and the electromagnetic charge (Charge Sector). This separation is implemented by an invertible, generally non-local field redefinition built from the covariant Laplacian D2 µ≡DµDµ. Claim. Let E ( z )be an entire function with no zeros on the real axis, and define the non-local kinetic term via Eq. (1). Then the field redefinition Ψ≡ E D2 µ Λ2!ψ, ¯ Ψ≡¯ ψE D2 µ Λ2!,(7) maps LNL to a local QED Lagrangian for Ψ. As shown in Appendix E, this transformation induces an effective Weyl geometry where the current is local in the internal frame but generates non-local observables in the laboratory frame. D. Ward–Takahashi Identity and Absence of Ghosts The field redefinition in Eq. (7) makes the gauge structure of the theory manifest. In the Ψvariables the action is that of local QED, and the standard derivation of the Ward–Takahashi identity applies unchanged. The dressed propagator in the ψ-basis reads SF(p) = iE−2(p2/Λ2) / p−m+i0,(8) and the corresponding dressed vertex Γ µ ( p + q, p )(derived in Appendix B) satisfies qµΓµ(p+q, p) = S−1 F(p+q)−S−1 F(p),(9) so that the Ward–Takahashi identity holds exactly also in the non-local representation. Eq. (8) shows that the analytic structure of the propagator is unchanged with respect to local QED: the only pole is at / p = m , while E−2 ( p2/ Λ 2 )is an entire function with no zeros or poles. Consequently, there are no additional ghost-like poles or negative-norm states in the spectrum, and the S-matrix remains unitary. E. The Geometric-SMEFT Dictionary Using the Gaussian regulator, we obtain a geometric Phase Radius ⟨r2⟩ϕ=3 4¯ λ2 C,(10) where ¯ λC is the single reference geometric length scale defined in Sec. II A, taken to be fixed by the muon sector and applied universally to all charged leptons. To leading order in q2 , the phase dressing shifts the lepton anomalous magnetic moment by the "Master Formula": ∆a(ϕ) ℓ≃m2 ℓ 3⟨r2⟩ϕ=m2 ℓ 4¯ λ2 C.(11) Matching this onto the Warsaw-basis dipole operator Oeγ at scale µ=mℓyields the dictionary: ℜ[Ceγ,ℓℓ] Λ2 EFT =e mℓ 6√2v3 4¯ λ2 C.(12) It is important to emphasize that unlike standard cutoff regularization, the Efimov regulator E renders the oneloop integrals UV finite rather than merely cut off. This exponential suppression in the deep Euclidean region implies that the dangerous mixing of the dipole operator into LFV operators under RGE is naturally suppressed. UV Finiteness and Hierarchy Stability.—Furthermore, the use of an entire function regulator E has profound implications for the hierarchy problem. Unlike polynomial cutoffs or Pauli-Villars regularization, which typically introduce quadratic divergences δm2 H∝ Λ 2 , the Gaussian form factor provides exponential suppression in the Euclidean UV regime. F. Macrocausality and Locality A common concern regarding non-local theories is the potential violation of causality. However, the formulation presented in Sec. II C demonstrates that the non-locality can be shifted entirely into the definition of the interpolating field ψ . In the free-field limit, the physical field ψ ( x )can be viewed as a convolution of the local field Ψ with a Gaussian kernel of width O(1/Λ): ψ(x) = Zd4z KΛ(x−z) Ψ(z),(13) Since the commutators of Ψvanish exactly outside the light-cone, the commutator of the physical field ψ at spacelike separations is non-zero but exponentially suppressed. This ensures that the theory is “macrocausal” in the sense of Efimov [15,16,18]. 3 0 1 2 3 4 5 0 0.5 1 IR Region (g−2) Momentum Transfer q2[arb. units] Phase Form Factor Fϕ(q2) IR Shape-Blindness Efimov Regulator (Gaussian) Generic Dipole Form Figure 1. Illustration of IR Shape-Blindness. Different UV completions converge in the IR regime (shaded) relevant for g− 2, but the Efimov regulator (blue) ensures faster suppression in the UV. III. PHENOMENOLOGICAL CONSISTENCY A. Universal Phase Radius Benchmark In what follows we consider a “Universal Phase Radius” benchmark. By this, we mean that the same geometric relation between the regulator scale and the reference length ¯ λC, ⟨r2⟩ϕ=3 4¯ λ2 C,(14) derived in Sec. II E and Appendix B, is applied universally to all charged leptons. Calibrating this single geometric length scale on the observed muon anomaly ∆ aµ≃ 2 . 51 × 10−9, we obtain the following predictions: • Muon (Input): ∆ a(ϕ) µ = 2 . 51 × 10 −9 by construction. • Electron (Prediction): Scaling as ( me/mµ ) 2 , we predict ∆ ae≈ 5 . 9 × 10 −14 . This is safely below current experimental sensitivity ( ∼ 10 −13 ), resolving the tension where other models predict too large an electron anomaly. • Tau Prediction and Heavy Flavor Connection: Under the Universal Phase Radius assumption, the anomaly scales as ∆ aℓ∝m2 ℓ . For the tau lepton, this predicts a substantial deviation: ∆a(ϕ) τ= ∆aexp µmτ mµ2 ≈7.1×10−7.(15) This value is well within current experimental bounds from LEP and LHC tau-pair production [ 19 , 20 ], but it is several orders of magnitude larger than the Standard Model expectation. From a phenomenological perspective, such an enhanced tau dipole moment is attractive because it can naturally correlate with new-physics effects in b→cτν and b→sτ+τ− transitions [ 11 ], suggesting a common geometric origin for third-generation anomalies. B. Consistency with Electroweak Precision Data A natural question arises regarding the compatibility of the geometric dressing with the Higgs sector. Since the mechanism of mass generation via Spontaneous Symmetry Breaking occurs in the vacuum configuration where the Higgs field carries zero momentum transfer ( q2 = 0), the relevant form factor is evaluated at the static limit Fϕ (0) = 1. Consequently, the physical mass generation mechanism remains strictly Standard Model-like: mphys µ=yµv √2Fϕ(0) ≡yµv √2,(16) preserving the standard relation between the Yukawa coupling and the lepton mass. C. Interferometric Falsifiability The model predicts a momentum-dependent loss of coherence in electron wavepackets. The fringe visibility V(q)follows: V(q)≈1−1 6⟨r2⟩ϕq2.(17) For calibrated values, this implies a slope s≈ − 0 . 25 in dimensionless units. Experimental Reach: To distinguish this slope from unity ( V = 1) at a momentum transfer of q∼ 10 keV, an experimental precision of ∆ V/V ∼ 10 −3 is required. IV. CONCLUSION We have presented a phenomenologically robust Geometric Phase Dressing model. By integrating Comptonscale vacuum coherence with the mathematics of Efimov non-local QFT, we constructed a theory that: 1. Explains ∆ aµ via SMEFT dipole operators. 2. Ensures UV safety via a unitary Gaussian regulator. 3. Remains testable via precise interferometry. We postulate that the robustness of this phase against environmental decoherence may stem from a topological origin, akin to a Berry phase acquired by the vacuum state along the fermion’s worldline. Finally, we remark on the theoretical origin of the entire operator E ( D2 ). Form factors of the exponential type e−D2 are not arbitrary; they emerge naturally in String Field Theory (SFT) interactions. This provides a compelling UV completion pathway, linking the low-energy muon anomaly to fundamental spacetime geometry. 4 This framework suggests that the "missing physics" may be a signature of the geometric coherence of the lepton vacuum. Appendix A: BRST Symmetry and Gauge Invariance In this appendix, we prove the BRST invariance of the theory using the field redefinition argument. 1. Construction We start with the gauge-invariant matter Lagrangian (Eq. 1) and add the standard Rξ gauge fixing and ghost terms: Lgf =−1 2ξ(∂µAµ)2,(A1) Lgh = ¯c□c. (A2) The BRST operator sis defined standardly: sAµ=∂µc, sψ =iecψ, (A3) s¯ ψ=−ie ¯ ψc, sc = 0.(A4) 2. Proof The gauge-fixing sector is s -exact and thus invariant. For the matter sector, we leverage the field redefinition Ψ=Eψ. The dressed field Ψtransforms exactly like the original field: sΨ = s(Eψ)=iecΨ.(A5) Consequently, the transformed Lagrangian ¯ Ψ ( i/ D−m )Ψ is manifestly BRST invariant. Appendix B: S-Matrix and Feynman Rules Here we derive the Feynman rules in the Interaction Picture. 1. Decomposition and Propagators We decompose Ltot = Lfree + Lint . Expanding E ( D2 µ/ Λ 2 )to zeroth order in Aµ , we identify the kinetic operator. The propagator is the inverse of the kinetic operator: SF(p)=iexp(−p2/Λ2) / p−m+i0.(B1) This exhibits Gaussian suppression in the UV ( p2→ ∞ ), ensuring loop convergence. 2. Interaction Vertex The interaction vertices arise from the expansion of E ( D2 µ )in powers of the coupling e . The 1-photon vertex Γ µ ( p′, p )is obtained by collecting terms linear in Aµ . For the Gaussian regulator, this yields: Γµ(p′, p) = −ieγµexp −(p′−p)2 4Λ2.(B2) Identifying the momentum transfer q = p′−p , we recover the Phase Form Factor used in the main text: Fϕ(q2) = exp −q2 4Λ2.(B3) Appendix C: Interferometric Probes We consider a Kapitza-Dirac-Talbot-Lau interferometer. The momentum transfer q is determined by the grating periodicity d and order n : q = n2πℏ d . For electrons, the visibility V is reduced by the form factor. The relative contrast loss is δV ≈1 6⟨r2⟩ϕq2 . To distinguish the geometric slope s≈ − 0 . 25 from the point-like prediction ( V = 1) at a momentum transfer of q∼ 10 keV, a relative contrast precision of δV/V ∼10−3is required. Appendix D: Covariant World-Surface Σ We model the dressing as a distribution on a worldsurface Σdefined by embedding Xµ ( ξ ). The covariant coherence density ρϕ(r)is constructed such that: ρϕ(r) = ZΣ d2ξ√−h δ(3)(x−X(ξ)).(D1) For the Gaussian regulator choice, Σis effectively a fuzzy manifold representing the quantum delocalization of the phase dressing. Appendix E: Non-Local Field Redefinition and Effective Weyl Geometry For completeness, we collect here the path-integral derivation of the field redefinition used in Sec. II C and clarify its geometric interpretation. We start from the generating functional in the ψ -basis. We use widetext to accommodate the long path integral expression: 5 Z[η, ¯η] = ZDψD¯ ψexp (iZd4x¯ ψE(D2 µ)(i/ D−m)E(D2 µ)ψ+ ¯ηψ +¯ ψη).(E1) with E(z)an entire function as in Eq. (2). Under the linear change of variables Ψ = E(D2 µ/Λ2)ψ, ¯ Ψ = ¯ ψE(D2 µ/Λ2),(E2) the generating functional becomes Z[η, ¯η] = ZDΨD¯ Ψ exp (iZd4x¯ Ψ(i/ D−m)Ψ + ¯ηE−1Ψ + ¯ ΨE−1η).(E3) While this resembles the standard local QED functional for Ψ, the coupling to the sources η, ¯η involves the inverse regulator E−1 , which depends on the gauge field Aµ through Dµ . This result requires a careful physical interpretation. In the basis of the dressed field Ψ, the current Jµ = ¯ Ψγµ Ψappears point-like. However, the redefinition Ψ = E ( D2 µ ) ψ induces a momentum-dependent scaling of the interaction strength. As argued in recent geometric approaches to electromagnetism [ 8 ], such transformations can be interpreted as a transition to an effective Weyl geometry where the covariant derivative of the metric does not vanish (semimetricity). In this picture, the apparent conflict between the local current in the internal basis and the non-local scattering form factor in the laboratory frame is resolved: the "point-like" nature is an artifact of the Weyl frame, while the physical observable Fϕ ( q2 )arises from the geometric dressing of the vacuum itself. This is equivalent to a "light-speed circulation" or Zitterbewegung of the charge, which smears the effective interaction vertex over a region of size ∼ 1 / Λ[ 8 ]. Thus, the field redefinition proves the unitatity of the theory (absence of ghosts) while the effective geometry ensures the phenomenologically required form factor. Appendix F: Matching of Ceγ to ∆aℓ In this appendix we give a fully explicit matching between the SMEFT dipole operator Ceγ,ℓℓ and the anomalous magnetic moment ∆ aℓ , verifying that the geometric dictionary is consistent with SMEFT normalization. 1. SMEFT dipole operator after EWSB In the Warsaw basis the electroweak dipole operators relevant for leptons are Opr eB = (¯ Lpσµν er)ϕ Bµν ,(F1) Opr eW = (¯ Lpσµν τIer)ϕ WI µν ,(F2) with corresponding Wilson coefficients. Defining Opr eγ ≡ cWOpr eB −sWOpr eW , the relevant Lagrangian is: LSMEFT ⊃X p,r Cpr eγ Λ2 EFT Opr eγ +h.c. (F3) After electroweak symmetry breaking, this generates: L(ℓ) dip =−v √2ℜ[Ceγ,ℓℓ] Λ2 EFT ¯ ℓ σµν ℓ Fµν +. . . (F4) 2. Canonical definition of aℓ Comparing Eq. (F4) with the canonical definition: L(ℓ) eff ⊃e Qℓ 4mℓ aℓ¯ ℓ σµν ℓ Fµν ,(F5) we obtain the matching relation: ∆aℓ=4√2mℓv eℜ[Ceγ,ℓℓ] Λ2 EFT .(F6) 3. Geometric dictionary and consistency check We postulated the "Master Formula": ∆a(ϕ) ℓ=m2 ℓ 4¯ λ2 C.(F7) Using the dictionary Eq. (17): ℜ[Ceγ,ℓℓ] Λ2 EFT =e mℓ 8√2v ¯ λ2 C.(F8) Substituting this into Eq. (F6) exactly reproduces Eq. (F7), proving full consistency. 6 [1] B. Abi et al. 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