Gauge Transformations Preserve Mass Generation in Reduced Quantum Electrodynamics

A theoretical study of reduced quantum electrodynamics, or RQED, shows that a fermion propagator computed in one covariant gauge can be transformed exactly into another gauge while preserving the gauge covariance required of physical observables. The researchers derived an all-orders expression for the charged-fermion propagator, expanded it through two loops for both massive and massless fermions, and showed that the transformation keeps the dynamically generated fermion pole mass and chiral condensate unchanged when gauge changes are handled consistently. Their results concern the theoretical RQED model with fermions confined to a lower-dimensional space and photons propagating in a higher-dimensional bulk; they do not constitute a laboratory measurement or a demonstration of a new phase in a material.

The work focuses on RQED_4,3, a setting in which fermions live in three spacetime dimensions while photons propagate in four. This geometry is associated with graphene-like condensed-matter systems whose low-energy quasiparticles can have relativistic, linear dispersion, but the present study is an analytical and numerical field-theory calculation rather than an experimental analysis of graphene. Reduced QED differs from ordinary four-dimensional QED in a consequential way: the photon propagator sampled by the fermions has a modified longitudinal component. Its effective gauge parameter is therefore ξ̃=(1−ε_e)ξ, where ε_e=(d_γ−d_e)/2 for photons in d_γ dimensions and fermions in d_e.

At the center of the analysis is the Landau–Khalatnikov–Fradkin transformation, a set of rules that connects Green functions evaluated in different covariant gauges. Developed originally for Abelian gauge theories, these transformations state in coordinate space that changing the gauge modifies a propagator through a known multiplicative factor involving a gauge-dependent coordinate-space function. For the three-dimensional fermion propagator considered here, the transformation reads S₃(x;ξ)=S₃(x;ξ₀) exp[−i(Δ̃₃(0)−Δ̃₃(x))]. After regulating the short-distance singularity with a cutoff x_min, the gauge dependence is encoded by ν=α(ξ−ξ₀)/(4π) and a power (x/x_min)^−2ν. This structure is exact in the sense that it follows from the gauge transformation itself, rather than being restricted to a finite perturbative expansion.

Beginning from the bare propagator in a reference gauge, the authors analytically transformed both its scalar Lorentz components and reconstructed corresponding expressions for the mass function and wave-function renormalization. The resulting momentum-space functions contain gamma functions, trigonometric terms and a gauge exponent. For a massive free fermion, these expressions recover the full dependence on the ratio p²/m², rather than only a high-momentum expansion. The authors then expanded the general result in powers of the coupling through order α², equivalent to ν², and in m²/p². The massless limit agreed with available perturbative calculations, while the massive case extends those earlier results to two loops.

A key outcome is that ξ₀=1/3 is the preferred reference gauge for RQED_4,3. In that gauge, the one-loop anomalous dimension of the massless fermion field vanishes because γ_ψ(α,ξ)=(2α/3)(1−3ξ)+O(α²). Consequently, the leading logarithmic part of the massless wave-function renormalization is absent at one loop. The choice plays a role analogous to that of Landau gauge in ordinary four-dimensional QED, where the one-loop massless anomalous dimension instead vanishes at ξ=0. The agreement between the transformed expression and direct perturbative results makes the choice ξ₀=1/3 more than a computational convenience: it links gauge covariance, perturbation theory and multiplicative renormalizability.

The researchers also used the expanded propagator to examine ultraviolet renormalization. The logarithmic divergences in the wave-function renormalization are separated through the renormalization constant Z₂, while the mass function does not exhibit the same logarithmic divergences. At leading-log accuracy, the renormalized massless wave-function renormalization exponentiates to F_R(p²/μ²;ξ)=(p²/μ²)^ν. The corresponding two-loop expression contains the expected terms through order ν² and agrees with the established massless result. The work therefore provides an exact gauge-mapping framework whose weak-coupling expansion is consistent with the ultraviolet structure of RQED.

The nonperturbative part of the study addresses dynamical chiral symmetry breaking, in which an interacting massless fermion can acquire a momentum-dependent mass even when its current mass is set to zero. The authors started with gap-equation solutions generated using three treatments of the fermion–photon vertex: the bare vertex, the Ball–Chiu vertex and an improved Curtis–Pennington vertex. For the initially massless theory in RQED_4,3, the gauge transformation assumes a particularly simple multiplicative form: both coordinate-space scalar functions acquire the same factor (x/x_min)^−2ν. After inverse Fourier transformation, the authors evaluated the resulting momentum integrals numerically. Integrable singularities at equal external and loop momenta were isolated analytically to make the computation stable.

As expected, the wave-function renormalization and momentum-dependent mass function themselves vary with the gauge parameter, because Green functions may carry gauge dependence. The physically relevant outputs behave differently. The Euclidean fermion pole mass remained essentially constant as the propagator was transformed between gauges, and the chiral fermion condensate also remained unchanged. The comparison showed a striking contrast with independently solving the truncated gap equation in each gauge: those separate solutions produced substantial gauge dependence, especially with the bare vertex. Transforming a solution from the reference gauge instead preserved the two physical observables. The near agreement between transformed Ball–Chiu and improved Curtis–Pennington results also suggests that the longitudinal vertex structure constrained by the Ward–Green–Takahashi identity dominates the gauge-covariant mapping, while additional transverse terms have a stronger effect on independently truncated gap-equation dynamics.

Overall, the study presents LKF transformations as a consistency tool for reduced quantum electrodynamics, linking exact gauge covariance to perturbative renormalization and nonperturbative mass generation. Its findings support using a single well-defined reference-gauge solution to analyze other covariant gauges without sacrificing the gauge invariance of the fermion pole mass or chiral condensate. The authors identify the next theoretical step as extending the transformation to the dressed fermion–photon interaction vertex, a three-point Green function that contains the fundamental interaction structure of the theory. The present conclusions remain within the modeled RQED_4,3 framework and the specified vertex approximations; they do not establish that the same quantitative behavior occurs in every graphene-like material or strongly correlated electronic system.

Subject of Research: Gauge transformations of the fermion propagator in reduced quantum electrodynamics

Article Title: Landau–Khalatnikov–Fradkin transformations in reduced quantum electrodynamics: perturbative and nonperturbative dynamics of the fermion propagator

Article References: Ashraf, A., Akram, F., Aslam, M. J., Rodríguez-Tzintzun, D., Bashir, A., & Albino, L. (2026). Landau–Khalatnikov–Fradkin transformations in reduced quantum electrodynamics: perturbative and nonperturbative dynamics of the fermion propagator. The European Physical Journal C, 86(10), Article 1194. https://doi.org/10.1140/epjc/s10052-026-16319-2

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DOI: 10.1140/epjc/s10052-026-16319-2

Keywords: reduced quantum electrodynamics, fermion propagator, gauge covariance, chiral symmetry breaking, dynamical mass generation, Landau–Khalatnikov–Fradkin transformations, two-loop calculation, wave-function renormalization, chiral condensate, graphene-like systems, Landau, Khalatnikov