Geometric Phase Dressing of the Standard Model Lepton Propagator: Effective Weyl Geometry, SMEFT Dipole Operators, and LHC Constraints

Geometric Phase Dressing of the Standard Model Lepton Propagator: Effective Weyl Geometry, SMEFT Dipole Operators, and LHC Constraints Damian Pikor and Paweł Kurzawski (Dated: November 28, 2025) The discrepancy between the Standard Model prediction and the experimental measurement of the muon anomalous magnetic moment, ∆ aµ , suggests potential new physics at the Compton scale. In this work, we investigate a theoretical framework where this anomaly 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. A crucial feature of this framework is the mapping to an effective Weyl geometry, where the electromagnetic current is local in the dressed frame but manifests as non-local interactions in the laboratory frame. To ensure the strict conservation of the electromagnetic current, we construct the interaction vertex using a non-local gauge link formalism, which satisfies the Ward-Takahashi Identity exactly. This approach naturally generates dimension-6 dipole operators in the Standard Model Effective Field Theory (SMEFT). Distinct from general non-local QED models that require multiple scale parameters, our framework relies on a single universal geometric phase scale. We calibrate this scale to the muon anomaly (Λ ≈ 3 . 2TeV) and analyze the tension with recent high-energy constraints from LHC Drell-Yan processes, demonstrating that the specific Gaussian form factor allows the model to remain viable within current exclusion limits. 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 a 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. II. THEORETICAL FRAMEWORK A. Non-Local Action and Regularization We postulate the following non-local Lagrangian, constructed to ensure Hermiticity and gauge invariance within the dressed frame formalism: LNL =−1 4F2 µν +¯ ψK D2 µ Λ2!(i/ D−m)K D2 µ Λ2!ψ, (1) where Dµ = ∂µ + ieAµ denotes the covariant derivative and / D = γµDµ is the Dirac operator. The operator K serves as the inverse of the regularization function, effectively delocalizing the interaction. To ensure the ultraviolet finiteness of loop integrals, the propagator must decay exponentially in the Euclidean region. Consequently, the kinetic operator in the Lagrangian must exhibit corresponding exponential growth. Adopting a Gaussian regulator, we define: K D2 µ Λ2!= exp +D2 µ 2Λ2!.(2) Here, Λrepresents the fundamental non-locality scale, related to the reference Compton wavelength via Λ = √2/¯ λC . The positive sign in the exponent of Eq. (2) is mathematically necessary to generate the required suppression factor exp ( −p2/ Λ 2 )in the dressed propagator derived in Appendix B. Note on Vacuum Polarization. While non-local interactions typically modify both the fermion and photon 2 propagators, in this work we focus on the geometric dressing of the matter fields. The contribution of non-local vacuum polarization to g− 2is of higher order in the expansion parameter and is suppressed relative to the vertex correction discussed here. B. Field Redefinition and Unitarity The physical consistency of this theory can be analyzed via a field redefinition. Let us define the "dressed" field variables: Ψ≡ K D2 µ Λ2!ψ, ¯ Ψ≡¯ ψK D2 µ Λ2!.(3) In terms of Ψ, the Lagrangian reduces to the standard local QED form. As shown by Efimov and Alebastrov [ 17 ], since K ( z )is an entire function with no zeros, this transformation does not introduce new poles in the finite complex plane. Consequently, the particle spectrum remains identical to that of the Standard Model (no ghosts), and the S-matrix is unitary in the subspace of physical states. C. Ward-Takahashi Identity Compliance In non-local theories, the definition of the interaction vertex is constrained by the Ward-Takahashi Identity (WTI): qµΓµ(p+q, p) = S−1 F(p+q)−S−1 F(p).(4) Given that the inverse propagator S−1 F ( p )grows exponentially in the UV (due to the K factors), the vertex Γ µ must exhibit corresponding growth to satisfy Eq. (4) . A "naive" vertex that decays in the UV would violate charge conservation. In Appendix B, we derive the explicit form of the vertex using the non-local gauge link formalism, demonstrating that it rigorously satisfies the WTI. D. SMEFT Matching In the low-energy limit ( q2≪ Λ 2 ), the non-local interactions can be expanded in a series of local operators. The leading contribution to the electromagnetic dipole moment comes from the non-commutativity of the covariant derivatives within the operator K ( D2 µ ). The matching to the SMEFT dipole operator Oeγ [12] yields: ℜ[Ceγ,ℓℓ] Λ2 EFT =e mℓ 6√2v3 4¯ λ2 C.(5) The numerical factor 3 / 4arises from the trace of the Gaussian regulator integral in the Euclidean loop expansion. 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. III. PHENOMENOLOGICAL ANALYSIS A. Universal Phase Radius Hypothesis We investigate the implications of a "Universal Phase Radius," assuming that the geometric scale ¯ λC is common to all charged leptons. Calibrating the model to the central value of the muon anomaly, ∆ aµ≈ 2 . 51 × 10 −9 , fixes the scale Λ. Under this assumption, the contributions to the anomalous magnetic moments scale with the lepton mass squared (m2 ℓ). This leads to the following predictions: • Electron: The predicted contribution is ∆ ae≈ 5 . 9 × 10 −14 . This value is below the current experimental sensitivity ( ∼ 10 −13 ) [ 7 ], ensuring compatibility with existing precision data. • Tau: The model predicts a significant enhancement for the tau lepton, ∆ aτ≈ 7 . 1 × 10 −7 . While this is consistent with current bounds from LEP and LHC [ 21 – 23 ], it suggests that future high-precision measurements of the tau sector could test this geometric hypothesis. B. Electroweak Symmetry Breaking The geometric dressing described here acts on the kinetic terms and gauge interactions. The mass generation mechanism via the Higgs Yukawa coupling occurs at zero momentum transfer ( q2 = 0). Since the form factor is normalized such that F (0) = 1, the physical masses of the leptons remain defined by their standard Yukawa relations, mℓ = yℓv/√2 . This ensures that the proposed dressing does not disrupt the Spontaneous Symmetry Breaking mechanism [20]. 3 C. Constraints from LHC Drell-Yan Data The "Universal Phase Radius" hypothesis implies that the non-local modification of the lepton propagator extends to the high-energy regime probed by the Large Hadron Collider (LHC). In the Drell-Yan process ( pp → ℓ+ℓ− ), the non-local form factors manifest as momentumdependent corrections to the dilepton invariant mass spectrum. For momentum transfers ˆs≪ Λ 2 , these corrections can be parameterized effectively as dimension-6 contact interactions in the SMEFT framework. Recent analyses of high-mass Drell-Yan tails by the ATLAS and CMS collaborations at √s = 13 TeV place stringent lower bounds on the scale of such non-local interactions. Specifically, for flavor-universal couplings, the non-locality scale is constrained to be Λ ≳ 3 . 0TeV [ 15 , 16 ]. Our calibration to the muon anomaly suggests a central value of Λ ≈ 3 . 2TeV, which lies in the tension region just beyond current exclusion limits. Crucially, however, experimental limits are typically derived assuming contact interactions (EFT limit, Λ → ∞ ) which lead to an enhancement of the cross-section (constructive interference) in the high-mass tail. In contrast, the Efimov form factor e−ˆs/Λ2 leads to a distinct phenomenological signature: a suppression (deficit) of the cross-section due to the destructive interference of the propagator in the s-channel. This qualitative difference implies that the bound Λ > 3 . 0TeV, derived for standard contact interactions models, may not apply strictly oneto-one to the Gaussian suppression case. This proximity suggests that the non-local geometric dressing scenario is falsifiable and could be definitively tested in the upcoming High-Luminosity LHC (HL-LHC) run, where the specific shape of the suppression would become statistically distinguishable from the Standard Model background. IV. CONCLUSION We have presented a consistent formulation of the geometric phase dressing model for the Standard Model leptons. By utilizing the Efimov formalism with a non-local gauge link, we resolved the theoretical tension between the UV regularization requirement and the Ward-Takahashi Identity. We demonstrated that the non-local field redefinition admits an interpretation in terms of an effective Weyl geometry. Phenomenologically, the model provides a viable explanation for the muon g− 2anomaly. The predicted scaling naturally suppresses the contribution to the electron magnetic moment while allowing for potentially observable effects in the tau sector. While the model operates at the threshold of current exclusion limits from LHC Drell-Yan searches, the specific Gaussian nature of the form factor offers a distinct phenomenological signature (cross-section suppression) that remains consistent with current high-mass tail data. The tension between the low-energy anomaly and high-energy bounds provides a clear roadmap for falsification at the HL-LHC. Finally, we suggest that the stability of this phase against environmental decoherence might stem from a topological origin, analogous to a Berry phase acquired by the vacuum state along the fermion’s worldline. Furthermore, form factors of the exponential type e−D2 naturally arise in String Field Theory (SFT) interactions, pointing to a possible UV completion linking the low-energy muon anomaly to fundamental spacetime geometry. Appendix A: BRST Symmetry The BRST invariance of the theory follows directly from the field redefinition argument presented in Section II.B. 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 K 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. 4), 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′ )is given by the difference quotient: Γµ(p, p′)=−ieγµe+p′2/Λ2−e+p2/Λ2 (p′2−p2)/Λ2.(B2) While this vertex grows exponentially in the ultraviolet, it is always contracted with external propagators in loop diagrams. The combined behavior ensures that the integrands are finite and well-behaved, SF ( p′ )Γ µSF ( p ) ∼ finite. 4 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 = K ( 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 Ψ, the current Jµ = ¯ Ψγµ Ψappears strictly local. However, the inverse transformation ψ = K−1 Ψintroduces the non-local regulator K−1. As discussed in recent geometric approaches to electromagnetism [ 8 ], such transformations can be viewed as a transition to an effective Weyl geometry where the covariant derivative of the metric does not vanish (semimetricity). In this picture, the apparent non-locality in the laboratory frame is an artifact of projecting a local interaction from the Weyl frame onto the standard Riemannian manifold of the observer. This provides a geometric rationale for the "point-like" behavior of the charge form factor FEM ≡ 1in the dressed basis, coexisting with the geometric phase form factor Fϕ ( q2 )in the physical frame. It is important to clarify that we treat this Weyl geometry as an effective description arising from the field dressing, rather than modifying the fundamental Einstein-Hilbert action. [1] B. Abi et al. (Muon g-2 Collaboration), Phys. Rev. Lett. 126, 141801 (2021). [2] Muon g-2 Collaboration, arXiv:2402.15410 (2024). [3] T. Aoyama et al., Phys. Rep. 887, 1 (2020). [4] A. Keshavarzi, K. S. Khaw, and T. Yoshioka, Nucl. Phys. B 975, 115675 (2022). [5] F. Jegerlehner and A. Nyffeler, Phys. Rep. 477, 1 (2009). [6] Particle Data Group, PTEP 2022, 083C01 (2022). [7] J. Aebischer et al., Phys. Rev. D 104, 115024 (2021). [8] J. Lindgren et al., J. Phys.: Conf. Ser. 2987, 012001 (2025). [9] G. Isidori, PoS BEAUTY2018, 060 (2018). [10] G. Isidori et al., JHEP 09, 060 (2021). [11] D. Guadagnoli and P. Koppenburg, arXiv:2207.01851. [12] B. Grzadkowski et al., JHEP 10, 085 (2010). [13] B. Henning et al., JHEP 01, 023 (2016). [14] A. Falkowski et al., Prog. Part. Nucl. Phys. 136, 104084 (2024). [15] ATLAS Collaboration, Phys. Lett. B 796, 68 (2019). [16] A. Biswas and N. Okada, Towards LHC physics with non-local Standard Model, Nucl. Phys. B 898, 79 (2015). [17] V. A. Alebastrov and G. V. Efimov, Commun. Math. Phys. 31, 1 (1973). [18] G. V. Efimov, Non-local Interactions of Quantized Fields, Nauka (1985). [19] L. Modesto and L. Rachwał, Nucl. Phys. B 889, 228 (2014). [20] L. Modesto, Phys. Rev. D 103, 076012 (2021). [21] A. Crivellin et al., PoS LHCP2024, 058 (2024). [22] A. Crivellin et al., SciPost Phys. 16, 048 (2023). [23] M. A. A. S. O. A. Khan et al., Phys. Rev. D 106, 095019 (2022). [24] M. Fabbrichesi et al., JHEP 05, 123 (2024). [25] A. Accioly et al., Phys. Rev. D 108, 055030 (2023). [26] H. Li and P. Wang, Eur. Phys. J. C 84, 654 (2024). [27] A. Crivellin and M. Hoferichter, JHEP 07, 135 (2021).