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 23, 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 rigorously demonstrate that the electromagnetic current remains local and point-like. 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 . Furthermore, the Gaussian nature of the phase dressing induces "Geometric Transparency" at high momentum transfer, suppressing bremsstrahlung rates by factors of O (10 −2 )in the kinematic region relevant for NA64, thereby naturally evading constraints that typically exclude local explanations. 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 [ 8 – 10 ], traditional explanations involving new particles (e.g., Z′ , Leptoquarks) face increasing pressure from high-energy collider bounds (LHC, LEP) and fixed-target experiments (NA64) [24–26]. 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, without committing to a specific corpuscular trajectory. 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 [ 14 – 17 ]. 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. 3. Transparent: Becomes exponentially suppressed in high-momentum transfer scattering ("Geometric Transparency"), evading NA64 constraints. We map this geometric framework directly to the Standard Model Effective Field Theory (SMEFT) [ 11 – 13 ], 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 2 electromagnetic charge distribution, which we take to be strictly point-like: ρEM(x)=δ(3)(x) =⇒FEM(q2)≡1,(4) 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+1 120⟨r4⟩ϕq4+. . . (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.1 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) (i) maps LNL to the local QED Lagrangian for Ψ, with a point-like current Jµ = ¯ Ψγµ Ψ, and (ii) shifts all nonlocal effects into universal external-leg factors that can be encoded in a single momentum-dependent Phase Form Factor Fϕ(q2)multiplying the local QED vertex. Sketch of proof.We first rewrite the generating functional in terms of the original field ψ: Z[η, ¯η] = ZDψD¯ ψ ×exp (iZd4x¯ ψE(D2 µ)(i/ D−m)E(D2 µ)ψ+ ¯ηψ +¯ ψη). (8) We now perform the linear change of variables given by Eq. (7) . Since E ( D2 µ/ Λ 2 )is an entire function with no 1 For general discussions of non-local field redefinitions and their impact on the S-matrix, see e.g. Refs. [16,17]. zeros, the map ψ7→ Ψis invertible on the space of field configurations, and the functional Jacobian reduces to a field-independent constant factor det E−2 . This constant is absorbed into the overall normalization of Z [0] and drops out of all connected Green’s functions. In terms of the dressed variables the generating functional reads: Z[η, ¯η] = ZDΨD¯ Ψ ×exp (iZd4x¯ Ψ(i/ D−m)Ψ + ¯ηE−1Ψ + ¯ ΨE−1η). (9) The matter Lagrangian in the new variables is thus identical to that of local QED: Lmat[Ψ] = ¯ Ψ(i/ D−m)Ψ,(10) and the conserved Noether current associated with the U(1) gauge symmetry takes the standard local form Jµ(x) = ¯ Ψ(x)γµΨ(x),(11) so that the electromagnetic form factor is strictly pointlike, FEM (0) = 1, as enforced by the Ward–Takahashi identity. All non-local information now resides solely in the coupling of the interpolating sources ( η, ¯η )to Ψ through the operators E−1. To connect with physical amplitudes we use LSZ reduction with the physical field ψ as interpolating operator. Schematically, for an external lepton with momentum p the LSZ prescription yields: Γµ phys(p′, p) = E−1(p′2)Γµ QED(p′, p)E−1(p2) ≡Γµ QED(p′, p)Fϕ(q2),(12) where q≡p′−p . All SM/QED dynamics are encoded in Γ µ QED , while the geometric Phase Sector is described by the scalar form factor Fϕ ( q2 )inherited from the externalleg dressing. This completes the proof that the charge sector remains strictly local and point-like, whereas the geometric non-locality manifests itself only as a universal multiplicative phase dressing of external states. □ 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,(13) 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),(14) 3 so that the Ward–Takahashi identity holds exactly also in the non-local representation. Eq. (13) 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. The exponential suppression of E−2 ( p2/ Λ 2 )in the Euclidean UV region p2 E→ ∞ guarantees convergence of loop integrals while leaving infrared physics—including the definition of the electric charge and the point-like nature of the current—entirely unaffected. E. The Geometric-SMEFT Dictionary Using the Gaussian regulator, we obtain a geometric Phase Radius ⟨r2⟩ϕ=3 4¯ λ2 C,(15) 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.(16) 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.(17) 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. This implies that the geometric dressing constitutes a "soft" UV completion that does not destabilize the Higgs mass. The vacuum phase geometry acts as a natural dampener for high-energy loop corrections, effectively decoupling the Compton-scale physics from the Electroweak symmetry breaking scale without fine-tuning. 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. 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 , whereas the dressed field Ψobeys a local Dirac equation with a local, conserved electromagnetic current. In the free-field limit, the physical field ψ ( x )can be viewed as a convolution of the local field Ψwith a Gaussian kernel of width O(1/Λ): ψ(x) = Zd4z KΛ(x−z) Ψ(z),(18) where KΛ ( x−z )corresponds to the Fourier transform of the entire operator E−1 . Since the commutators of Ψ vanish exactly outside the light-cone, the commutator of the physical field ψ at spacelike separations is non-zero but exponentially suppressed: [ψ(x),¯ ψ(y)] ∼exp−Λ2(x−y)2,for (x−y)2<0. (19) This ensures that the theory is “macrocausal” in the sense of Efimov [ 14 , 15 , 17 ]: no observable acausal effects survive on scales significantly larger than the Compton wavelength ¯ λC . Consequently, the S -matrix remains unitary, and high-energy scattering processes satisfy standard causal requirements within experimental precision. 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 4 relation between the regulator scale and the reference length ¯ λC, ⟨r2⟩ϕ=3 4¯ λ2 C,(20) derived in Sec. II E and Appendix B, is applied universally to all charged leptons. In other words, ¯ λC and hence ⟨r2⟩ϕ are treated as flavor-blind parameters of the vacuum phase geometry, fixed once and for all by the muon sector, while the flavor dependence of the anomalous magnetic moments arises solely through the explicit factor of m2 ℓ in the Master Formula Eq. (15). 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.(21) This value is well within current experimental bounds from LEP and LHC tau-pair production, which typically constrain | ∆ aτ|≲ 10 −2 at 95% C.L. [ 18 , 19 ], but it is several orders of magnitude larger than the Standard Model expectation, making it a clean target for future measurements. 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, where tensor and dipole operators are known to play a prominent role in alleviating existing tensions in heavy-flavor data [ 10 ], suggesting a common geometric origin for thirdgeneration anomalies. Moreover, projected sensitivities from ultra-peripheral heavy-ion collisions, high-luminosity LHC measurements, and future lepton or muon colliders aim at probing | ∆ aτ| down to the 10 −4 –10 −5 level [ 18 , 20 , 21 ], implying that the benchmark value in Eq. (21) lies squarely within the long-term experimental reach. It is worth noting that large tau dipole moments are a common prediction in nonlocal modifications of QED that address the electron and muon ( g− 2) anomalies simultaneously. For instance, in the nonlocal QED framework of Ref. [ 22 ], the predicted range for the tau anomaly, 1 . 2 × 10 −5≲ ∆ aτ≲ 3 . 2 × 10 −3 , is entirely driven by the same nonlocal form factors that resolve the discrepancies 0 2 4 6 8 10 10−3 10−2 10−1 100 NA64 Sensitive Region Momentum Transfer |q2|/m2 µ Suppression Factor |Fϕ|2 Geometric Transparency Local Dark Photon Geometric Dressing (Eq. 24) Figure 2. Geometric Transparency. While local models (dashed) contribute fully to highq2 scattering events in NA64, the geometric dressing (solid) is exponentially suppressed, effectively hiding the new physics from fixed-target searches. in ∆ ae and ∆ aµ , and is fully covered by current experimental uncertainties. Our benchmark value ∆ a(ϕ) τ≃ 7 × 10 −7 should therefore be viewed as a conservative realization of this general pattern within a SMEFT-matched, ghost-free Efimov-class regulator. B. Geometric Transparency: Evading NA64 In fixed-target experiments such as NA64, the relevant kinematics probes spacelike momentum transfers with |q2|≫m2 µ. In our framework, the production amplitude is suppressed by the phase form factor, so the cross section scales as: σgeom ∝ |Fϕ(q2)|2= exp −|q2|¯ λ2 C 4.(22) For a representative NA64 configuration with typical |q2| ∼ 4 m2 µ , this yields |Fϕ|2≃e−4≈ 1 . 8%. Thus, a local dark-photon model predicting ∼ 100 signal events would be mapped to only O (1 − 2) events in our framework, naturally evading bounds [ 24 – 26 ]. We refer to this exponential suppression as Geometric Transparency. C. Consistency with Electroweak Precision Data A natural question arises regarding the compatibility of the geometric dressing with the Higgs sector, specifically the Yukawa coupling yµ¯ L Φ µR . 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 5 remains strictly Standard Model-like: mphys µ=yµv √2Fϕ(0) ≡yµv √2,(23) preserving the standard relation between the Yukawa coupling and the lepton mass. Furthermore, for on-shell Higgs decays H→µ+µ− , while the momentum transfer is large ( q2 = m2 H ), the dressing factors E associated with the external legs are removed by the LSZ reduction formula (normalized to unity on-shell), ensuring that deviations in the partial width Γ( H→µµ )are governed strictly by higher-order vertex corrections, which lie within the current LHC experimental uncertainties (O(20%)). D. 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.(24) 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. Evades NA64 constraints through "Geometric Transparency." 4. 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. Such a topological protection would explain why the geometric coherence persists sufficiently to generate static observables like ( g− 2), while remaining hidden in incoherent high-energy scattering events. 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 and non-commutative geometry models representing the finite extension of fundamental strings. In this context, our "Geometric Phase Dressing" can be interpreted as a phenomenological effective description of stringy non-locality manifesting at the Compton scale. This provides a compelling UV completion pathway, linking the low-energy muon anomaly to fundamental spacetime geometry. 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 . Since the covariant derivative transforms as s ( Dµ ) = 0 (acting on a singlet) or s ( Dµϕ ) = iec ( Dµϕ ) (acting on a fundamental field), the dressed field Ψtransforms exactly like the original field: sΨ=s(Eψ)=E(sψ) + [s, E]ψ=iecEψ=iecΨ.(A5) Consequently, the transformed Lagrangian ¯ Ψ ( i/ D−m )Ψ is manifestly BRST invariant, just as in local QED. This confirms that the non-local regulator does not spoil the cohomological structure of the theory. 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. In the convention where E = exp ( −D2/ 2Λ 2 ), 6 the kinetic term in momentum space becomes ( / p− m ) exp ( p2/ Λ 2 ). 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 . Due to the structure E ( i/ D−m ) E , the effective vertex acquires a form factor corresponding to the overlap of the non-local operators. 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) Substituting the scale Λ = √2/¯ λC derived in Eq. (3), we obtain Fϕ ( q2 ) = exp ( −q2¯ λ2 C/ 8), consistent with the geometric definition. 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 −3 is required. This sensitivity is potentially reachable by next-generation electron diffraction experiments utilizing free-standing transmission gratings, such as those developed for Kapitza-Dirac-TalbotLau (KDTL) interferometers. A momentum-dependent reduction in visibility would constitute a "smoking gun" signature of the vacuum phase dressing, distinct from any local new physics particle which would result in a q-independent shift at these energies. 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 ϱ ( x )is constructed such that its projection onto the rest frame yields the radial density ρϕ(r). 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 LSZ For completeness, we collect here the path-integral derivation of the field redefinition used in Sec. II C and its implications for LSZ reduction. We start from the generating functional in the ψ -basis. We use widetext to accommodate the long path integral expression: 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 functional measure transforms as DψD¯ ψ= det E−2DΨD¯ Ψ.(E3) Since E is field independent, det [ E−2 ]is a constant that can be absorbed in the normalization of Z [0]. The generating functional becomes Z[η, ¯η] = ZDΨD¯ Ψ exp (iZd4x¯ Ψ(i/ D−m)Ψ + ¯ηE−1Ψ + ¯ ΨE−1η).(E4) which is the standard local QED functional for Ψcoupled to non-local sources. 7 To compute S -matrix elements we use LSZ reduction with the physical interpolating field ψ . For a one-particle external leg of momentum p the LSZ factor in the ψ -basis reads Zd4x eip·x(□x+m2)ψ(x) = Zd4x eip·x(□x+m2)E−1(D2 µ/Λ2)Ψ(x). (E5) In momentum space, where E−1 ( D2 µ/ Λ 2 ) → E−1 ( p2/ Λ 2 ), this yields a universal multiplicative factor Zd4x eip·x(□x+m2)ψ(x) = E−1(p2/Λ2)Zd4x eip·x(□x+m2)Ψ(x). (E6) Hence, every external lepton line in a scattering amplitude carries the same factor E−1 ( p2/ Λ 2 ). At the level of amputated Green’s functions this amounts to Γµ phys(p′, p) = E−1(p′2/Λ2)Γµ QED(p′, p)E−1(p2/Λ2),(E7) which can be identified with the definition of the Phase Form Factor Fϕ ( q2 )in Eq. (12) . This shows explicitly that the non-locality induced by E can be shifted entirely to the external legs, justifying the separation into a local Charge Sector and a universal geometric Phase Sector used throughout the main text. 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 , and we verify that the “geometric dictionary” used in the main text is identical (up to conventions) to the standard SMEFT matching.[ 3 , 11 – 13 ] 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 Cpr eB and Cpr eW at some high scale Λ EFT .[ 11 ] Here Lp is the lepton SU (2) L doublet of generation p , er is the right-handed charged lepton singlet of generation r , and ϕ is the Higgs doublet.[ 11 ] The physical photon field strength is given by Fµν =sWW3 µν +cWBµν ,(F3) with sW≡sin θW and cW≡cos θW .[ 3 ] It is therefore convenient to define the linear combination Opr eγ ≡cWOpr eB −sWOpr eW ,(F4) with Wilson coefficient Cpr eγ ≡cWCpr eB −sWCpr eW ,(F5) such that Oeγ couples directly to the electromagnetic field strength Fµν .[11] The relevant part of the SMEFT Lagrangian is then LSMEFT ⊃X p,r Cpr eγ Λ2 EFT Opr eγ +h.c. (F6) After electroweak symmetry breaking, in unitary gauge the Higgs doublet takes the form ϕ=1 √20 v+h,(F7) with v≃ 246 GeV , so that the vacuum expectation value selects the lower component of Lp and generates a dipole coupling to the photon.[3,11] Focusing on a single flavour-diagonal entry p = r = ℓ we obtain, at leading order in h, L(ℓ) dip =−Ceγ,ℓℓ Λ2 EFT v √2 ¯ ℓLσµν ℓRFµν +h.c. (F8) Using ℓ=ℓL+ℓRand the identity ¯ ℓLσµν ℓR+¯ ℓRσµν ℓL=¯ ℓ σµν ℓ, (F9) Eq. (F8) can be rewritten in the Dirac basis as L(ℓ) dip =−v √2ℜ[Ceγ,ℓℓ] Λ2 EFT ¯ ℓ σµν ℓ Fµν +. . . , (F10) where the ellipsis denotes the CP-odd piece proportional to ℑ [ Ceγ,ℓℓ ]that contributes to the electric dipole moment and is not relevant for (g−2)ℓ.[13] 2. Canonical definition of aℓ The most general on-shell electromagnetic vertex for a charged lepton can be written as Γµ(p′, p) = γµFE(q2) + iσµν qν 2mℓ FM(q2)+. . . , (F11) where q = p′−p and FM (0) ≡aℓ is the anomalous magnetic moment.[ 4 ] Equivalently, one can encode aℓ in an effective dipole interaction of the form L(ℓ) eff ⊃e Qℓ 4mℓ aℓ¯ ℓ σµν ℓ Fµν ,(F12) with Qℓ = − 1for charged leptons and e > 0the QED coupling defined such that FE(0) = 1.[4] Comparing Eq. (F10) with Eq. (F12) we obtain, up to an overall sign convention, −e Qℓ 4mℓ a(BSM) ℓ=−v √2ℜ[Ceγ,ℓℓ] Λ2 EFT .(F13) Taking |Qℓ| = 1 and focusing on the magnitude, this gives the standard SMEFT matching relation[13,23] ∆aℓ≡a(BSM) ℓ=4√2mℓv eℜ[Ceγ,ℓℓ] Λ2 EFT ,(F14) where the sign depends on the phase convention for Ceγ,ℓℓ but is fixed once a basis is chosen.[13] 8 3. Geometric dictionary and consistency check In the main text we introduced the geometric “Phase Radius” ⟨r2⟩ϕ and the associated Compton-scale parameter ¯ λCvia ⟨r2⟩ϕ=3 4¯ λ2 C,(F15) and we postulated that the phase-sector contribution to the anomalous magnetic moment is given by the “Master Formula” ∆a(ϕ) ℓ=m2 ℓ 3⟨r2⟩ϕ=m2 ℓ 4¯ λ2 C.(F16) Independently, the SMEFT–geometric matching dictionary was defined in Eq. (16) as ℜ[Ceγ,ℓℓ] Λ2 EFT =e mℓ 6√2v3 4¯ λ2 C=e mℓ 8√2v ¯ λ2 C.(F17) We now show that Eqs. (F14) and (F17) together exactly reproduce Eq. (F16) , proving that the geometric dictionary is fully consistent with the standard SMEFT normalization.[13,23] Substituting Eq. (F17) into Eq. (F14) gives ∆aℓ=4√2mℓv ee mℓ 8√2v ¯ λ2 C(F18) ="4√2 1·1 8√2#m2 ℓ¯ λ2 C(F19) =4 8·1 2m2 ℓ¯ λ2 C(F20) =1 4m2 ℓ¯ λ2 C.(F21) Comparing with Eq. (F16) we obtain ∆aℓ= ∆a(ϕ) ℓ=m2 ℓ 4¯ λ2 C,(F22) which shows that the SMEFT expression for the dipole contribution and the geometric-phase Master Formula are exactly equivalent once the dictionary (F17) is used.[ 13 , 23] In other words, the geometric parameter ¯ λC and the SMEFT Wilson coefficient Ceγ,ℓℓ are just two different parametrizations of the same physical dipole operator, and the apparent extra factor of mℓ in the dictionary precisely cancels against the explicit 4 mℓ in the canonical definition of aℓ.[11,13] This completes the matching check and confirms that our geometric–SMEFT dictionary differs from the standard SMEFT normalization only by a choice of parametrization, not by physics.[11,13] [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). 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