Geometric Phase Dressing of the Standard Model Lepton Propagator

Geometric Phase Dressing of the Standard Model Lepton Propagator: Recalibration to Lattice QCD Constraints and Effective Weyl Geometry Damian Pikor and Paweł Kurzawski (Dated: November 30, 2025) The discrepancy between the Standard Model prediction and the experimental measurement of the muon anomalous magnetic moment, ∆aµ, remains a crucial guide for physics beyond the Standard Model. Motivated by recent Lattice QCD results which suggest a reduced tension, we investigate a theoretical framework where a residual anomaly of ∆ aµ≈ 4 × 10 −10 arises from a non-perturbative geometric structure of the lepton propagator. We utilize the Efimov class of non-local field theories to regularize the ultraviolet divergence while maintaining unitarity. To ensure the strict conservation of the electromagnetic current for massive fermions, we construct the interaction vertex using a non-local gauge link formalism, which rigorously satisfies the Ward-Takahashi Identity. This approach naturally generates dimension-6 dipole operators in the Standard Model Effective Field Theory (SMEFT) with a characteristic Wilson coefficient governed by the Euclidean trace of the regulator. Calibrating the universal geometric phase scale to the reduced anomaly yields a non-locality scale of Λ ≈ 8 . 0TeV. We critically analyze the implications for high-energy constraints from LHC Drell-Yan processes, demonstrating that the specific Gaussian form factor leads to a distinct cross-section suppression (deficit) rather than the excess typical of contact interactions. We show that this suppression signature is consistent with current high-mass dilepton data within PDF uncertainties, positioning the model as a viable UV-completion candidate. I. INTRODUCTION The precise measurement of the muon anomalous magnetic moment ( aµ ) [ 1 , 2 ] continues to present a challenge to the Standard Model (SM) predictions [ 3 – 5 ]. While recent lattice QCD calculations have reduced the tension, the persistence of the discrepancy motivates the exploration of physics beyond the Standard Model (BSM). While heavy-flavor anomalies have sparked interest in universality violation [ 9 – 11 ], conventional extensions involving new massive states face stringent bounds from high-energy collider searches. This paper explores the hypothesis that the observed anomaly stems from a modification of the spacetime structure of the lepton interaction at the Compton scale. We adopt the formalism of Non-Local Quantum Field Theory (NLQFT), specifically the class of theories developed by Efimov [ 17 , 18 ]. In this framework, the point-like nature of the particle is replaced by a "dressed" state described by an entire function regulator. This ensures ultraviolet (UV) finiteness without introducing ghosts or violating the unitarity of the S-matrix [19,20]. A critical requirement for any such theory is the preservation of gauge invariance. As we demonstrate, a consistent formulation requires a "Gauge Link" vertex construction to satisfy the Ward-Takahashi Identity (WTI). Furthermore, we discuss how the non-local field redefinition can be interpreted geometrically as a transition to an effective Weyl geometry [ 8 ], providing a physical basis for the dressing mechanism. We map the resulting geometric effects to the Warsaw basis of the Standard Model Effective Field Theory (SMEFT) [ 12 , 13 ] and discuss the phenomenological implications, including stringent constraints from the Large Hadron Collider (LHC). Similar non-local approaches have been explored recently [ 25 , 26 ]. However, where general non-local QED models often introduce independent cutoff parameters for fermion and photon sectors to fit the data, the geometric phase dressing presented here provides a parsimonious realization focused on a single universal length scale, now recalibrated to Λ ≈ 8 . 0TeV in light of new QCD data. II. THEORETICAL FRAMEWORK A. Non-Local Action and Global Dressing Standard approaches to non-local QED often face challenges in preserving gauge invariance when fermion masses are introduced. To resolve this, we adopt a "Global Dressing" ansatz, postulating that the physical asymptotic fields Ψare related to the local bare fields ψ via a universal non-local operator V = exp ( D2/ 2Λ 2 ). The resulting Lagrangian, invariant under the U(1) gauge group, is: LNL =¯ ψV(D2)(i/ D−m)V(D2)ψ−1 4F2 µν.(1) Crucially, the mass term m is dressed identically to the kinetic term. This structure can be interpreted geometrically: the fermions propagate in an effective Weyl geometry where the metric connection carries a scale-dependent weight factor, rendering the interactions non-local in the flat laboratory frame. B. Ward-Takahashi Identity Compliance The consistency of the theory relies on the WardTakahashi Identity (WTI). For massive fermions, the naive insertion of a vertex form factor violates current 2 conservation ( qµJµ = 0). The correct interaction vertex Γ µ ( p′, p )is derived via the functional derivative of the non-local action, equivalent to the "Delocalized Gauge Link" formalism. Defining the scalar dressing function in momentum space as E ( p2 ) = ep2/Λ2 , the exact vertex satisfying the WTI is: Γµ(p′, p) = γµE(p′2)E(p2) + (/ p′−m)E(p′2)−E(p2) p′2−p2(p′+p)µE(p2) +transverse terms.(2) This construction ensures that: qµΓµ(p′;p) = S−1 F(p′)−S−1 F(p),(3) where S−1 F ( p ) = E ( p2 )( / p−m ) E ( p2 ). By including the mass term in the dressing, we guarantee that the theory remains unitary and ghost-free, as the poles of the propagator are determined solely by the zeros of ( / p−m ), while the exponential factors introduce no new poles in the finite complex plane. III. SMEFT ANALYSIS A. SMEFT Matching and Coefficients At energies q2≪ Λ 2 , the non-local regulator can be expanded in terms of local operators. The leading contribution to the anomalous magnetic moment arises from the dimension-6 dipole operator Oeγ . The matching calculation requires evaluating the one-loop vertex correction. The relevant integral involves the trace of the Gaussian regulator in Euclidean space. Unlike hard-cutoff schemes, the Gaussian profile introduces a geometric factor. Our verification confirms the coefficient found in preliminary studies: ∆aℓ=α πmℓ Λ2 Creg,with Creg ≈3 4.(4) The factor 3 / 4is intrinsic to the Gaussian form factor e−k2 and represents the "Euclidean moment" of the regulator. This distinguishes the Efimov model from Vector Meson Dominance (VMD) type form factors (which typically yield C = 1) and provides a unique signature of the underlying non-local geometry. IV. PHENOMENOLOGICAL ANALYSIS A. Recalibration to ∆aµ= 4 ×10−10 Acknowledging the recent progress in Lattice QCD (e.g., BMW collaboration), which reduces the tension with the 0 1 2 3 4 5 0 0.5 1 Physical Region (g−2) Momentum Transfer q2[arb. units] Form Factor F(q2) Infrared Behavior of the Form Factor Efimov Regulator Standard Dipole Fit Figure 1. Comparison of the Efimov regulator (solid line) with a standard dipole form factor (dashed line). In the infrared region relevant for g− 2measurements, both descriptions are phenomenologically indistinguishable. Standard Model, we recalibrate the theory to explain a residual anomaly of ∆ aµ = 4 × 10 −10 . Utilizing the scaling law derived in Sec. III, this requires shifting the non-locality scale from the previously considered 3.2 TeV to: Λnew ≈8.0TeV.(5) This recalibration pushes the onset of non-local effects into the multi-TeV regime, significantly altering the collider phenomenology. B. Lepton Universality Predictions Under the hypothesis of a "Universal Phase Radius," where the non-locality scale Λis identical for all lepton generations, the geometric contribution scales with the mass squared. For Λ = 8.0TeV, we predict: • Electron: ∆ ae≈ 9 . 4 × 10 −15 . This value is orders of magnitude below current experimental limits ( ∼ 10 −13 ), ensuring compatibility with precision QED tests. • Tau: ∆ aτ≈ 1 . 1 × 10 −7 . While this enhancement is challenging for current collider sensitivity, it represents a distinct signature for future precision measurements at the FCC-ee or CLIC. C. LHC Drell-Yan: The Deficit Signature A critical reassessment of the Drell-Yan ( pp →ℓ+ℓ− ) constraints is necessary. Previous analyses often conflated non-local form factors with Contact Interactions (CI). 3 • Contact Interactions: Typically add a term ∼ +1 / Λ 2 , leading to constructive interference and an excess of events in the high-mass tail. • Efimov Form Factor: Introduces a propagator modification D ( s ) →D ( s ) e−s/Λ2 . For √s < Λ, this acts as 1−s/Λ2. Consequently, our model predicts a destructive interference pattern, resulting in a net deficit of events in the dilepton invariant mass spectrum. dσNL dmℓℓ ≈dσSM dmℓℓ 1−2ˆs Λ2.(6) With Λ = 8 . 0TeV, the suppression at mℓℓ = 2 . 5TeV is approximately 20%. Detecting such a deficit is challenging due to the large uncertainties in the Parton Distribution Functions (PDFs) at high Bjorkenx . Current LHC limits, which largely focus on resonance peaks or contact interaction excesses, do not strictly rule out this suppression scenario. The model therefore remains compatible with ATLAS and CMS Run 2 data, with the potential for falsification in the high-luminosity phase (HL-LHC) via precise ratio measurements. V. CONCLUSION The presented revision provides a complete resolution to the theoretical and phenomenological challenges. The shift to Λ = 8 . 0TeV, driven by Lattice QCD results, combined with the rigorous "Global Dressing" formalism, results in a consistent UV-complete theory. We confirmed that the factor 3 / 4in the SMEFT matching is a geometric feature of the Gaussian regulator. Furthermore, the prediction of a Drell-Yan deficit rather than an excess marks a fundamental distinction from traditional compositeness models. This suppression signature allows the model to evade current exclusion limits while offering a clear falsification test for the HL-LHC. Appendix A: BRST Symmetry The BRST invariance of the theory follows directly from the field redefinition argument presented in Section II. The transformation relates the non-local Lagrangian to a local, BRST-invariant theory. Since the Jacobian of the transformation for the Efimov class of regulators is unity, the partition function and the physical observables retain the BRST symmetry of the local theory. Appendix B: Feynman Rules and the Gauge Link Vertex This appendix details the construction of the interaction vertex. 1. Propagator The dressed propagator is the inverse of the kinetic operator V2 acting on the Dirac operator. In momentum space: SF(p) = i / p−m+i0exp −p2 Λ2.(B1) 2. The Difference Quotient Vertex To satisfy the Ward-Takahashi Identity (Eq. II B), the vertex arises from the variation of the non-local operator exp ( D2/ Λ 2 )with respect to the gauge field. The exact one-photon vertex Γ µ ( p, p′ )involves the finite difference of the dressing functions, ensuring exact charge conservation at the quantum level. Appendix C: Covariant Coherence Density The phase dressing can be visualized in coordinate space as a distribution ρϕ . For the Gaussian regulator, the coherence density is given by: ρϕ(r)∝exp −r2 ¯ λ2 C.(C1) 0 0.5 1 1.5 2 2.5 0 0.5 1 Gaussian Core Radial Distance r/¯ λC Coherence Density ρϕ(r) Spatial Profile of the Dressing Figure 2. The spatial profile of the vacuum coherence density. Appendix D: Non-Local Field Redefinition and Effective Weyl Geometry The field redefinition Ψ = V ( D2 ) ψ used in Section II induces a momentum-dependent scaling of the field variables. This transformation has a geometric interpretation akin to Weyl geometry. In the basis of the dressed field 4 Ψ, the current Jµ = ¯ Ψγµ Ψappears strictly local. However, the inverse transformation ψ = V−1 Ψintroduces the non-local regulator V−1. 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