Pseudo Generalized QED (PGQED)

• PGQED is an extension of QED that incorporates higher-derivative Podolsky terms and confines Dirac fermions to a (2+1)D plane.
• It modifies the photon propagator with nonlocal, square-root momentum dependence, yielding screened Coulomb interactions relevant for planar materials.
• Schwinger–Dyson analyses in PGQED reveal dynamical mass generation and chiral symmetry breaking, establishing critical coupling and phase transitions.

Pseudo Generalized Quantum Electrodynamics (PGQED) refers to a class of extensions and dimensional reductions of quantum electrodynamics (QED) incorporating higher-derivative terms—specifically by the inclusion of Podolsky’s term—and applying dimensional reduction techniques. PGQED models are formulated by confining Dirac currents to (2+1)D planes while allowing gauge fields to propagate in the full (3+1)D bulk. This formalism provides nonlocal modifications to QED, with distinct dispersion relations and screened Coulomb interactions. The methodology is suited for analyzing strongly coupled planar Dirac fermion systems, such as graphene, with a focus on dynamical mass generation, chiral symmetry breaking, and ultraviolet regularization.

1. Formulation and Dimensional Reduction

PGQED originates from Podolsky quantum electrodynamics, which augments the standard Maxwell Lagrangian with a higher-order derivative term, parameterized by a mass scale μ (μ = 1/a, where a is the Podolsky parameter). The full (3+1)D Lagrangian is reduced such that matter currents (Dirac fermions) are strictly confined to a spatial plane, while the gauge field retains bulk propagation. The effective Lagrangian in (2+1)D for PGQED is given by: $$\mathcal{L}_\text{PGQED} = \frac{1}{4}\,\widetilde{F}_{\mu\nu} K_E[\Box] \widetilde{F}^{\mu\nu} + \bar{\psi}(i \not{\partial} - m)\psi + e\, \bar{\psi} \gamma^\mu \psi\, \widetilde{A}_\mu + \frac{\xi}{2} (\partial_\mu \widetilde{A}^\mu) K_E[\Box] (\partial_\nu \widetilde{A}^\nu)$$ where $\widetilde{A}_\mu$ and $\widetilde{F}_{\mu\nu}$ refer to gauge fields and field strengths in planar coordinates ($\mu=0,1,2$), and $K_E[\Box]$ is a nonlocal pseudo-differential operator extracted from the reduction of the Podolsky term: $$K_E[\Box] = \frac{2(-\Box + \mu^2)}{(-\Box + \mu^2)(-\Box)^{1/2} - (-\Box)[(-\Box + \mu^2)]^{1/2}}$$ The ultraviolet cutoff $\Lambda$ is introduced in momentum space to regulate loop integrals arising in nonperturbative analysis.

2. Gauge Field Propagator and Screened Interaction

PGQED modifies the photon sector substantially. The planar reduction of the higher-derivative term yields a gauge propagator with nontrivial, square-root momentum dependence: $G_{\text{PGQED}}(k) \sim \frac{K_E[k^2]}{k^2}$ In the static limit, the electron–electron interaction potential obeys: $V(r) \sim \frac{1-\exp(-r/a)}{r}$ This potential interpolates between the unscreened Coulomb interaction ($1/r$) at long distances and a regularized, finite value near the origin. The Podolsky parameter μ sets the finite screening length—a physical significance in condensed matter systems such as graphene, where the interaction is governed not only by dimensional constraints but also by the dielectric environment and lattice cutoff.

3. Schwinger–Dyson Equations and Dynamical Mass Generation

Critical analysis of strongly coupled PGQED is enabled via the Schwinger–Dyson formalism for the fermion propagator. The full propagator in Euclidean space admits a self-energy decomposition: $S_F^{-1}(p) = -A(p)\, \not{p} + \Sigma(p)$ In absence of a bare mass, dynamical mass generation is probed via a nonlinear integral equation for $\Sigma(p)$, incorporating the nonlocal structure of the gauge propagator. Under the rainbow (ladder) approximation and subsequent refinements (unquenched vacuum polarization corrections), an analytic gap equation is derived. The solution demonstrates chiral symmetry breaking once the effective coupling exceeds a critical value $\alpha_c(\mu)$: $$\left|M\right| \approx \frac{\Lambda \mu \exp(-\frac{\pi}{2\alpha})}{\Lambda + \sqrt{\Lambda^2 + \mu^2}}$$ The dynamical mass opens a gap in the Dirac spectrum, transforming massless fermions to massive excitations.

4. Analytical and Numerical Determination of Phase Structure

PGQED admits exact and approximate solutions for the critical parameters governing symmetry breaking:

• Analytical estimation of the critical coupling:

$\alpha_c(\mu) = \frac{\pi}{8} H(\Lambda,\mu)$

with $H(\Lambda,\mu)$ an increasing function for decreasing μ, signifying stronger screening demands a higher coupling for mass generation.

• Critical number of fermion flavors $N_c(\mu)$: Derived in the unquenched approximation, N_c mirrors values found in planar QED, converging to $N_c = 32/\pi^2$ under specific limits.
• The mass function $\Sigma(p)$ exhibits Miransky scaling near the critical point:

$$\Sigma(p) \propto p^{\gamma_-}, \quad \gamma_- = -\frac{1}{2} \pm \frac{1}{2}\sqrt{1-\frac{\alpha}{\alpha_c}}$$

Numerical methods via momentum discretization confirm the analytical phase structure, supporting the nontrivial onset of mass generation and position of the quantum phase transition in PGQED.

5. Validity Regime and Role of Ultraviolet Cutoff

PGQED’s predictive region is bounded by the ultraviolet cutoff $\Lambda$ and the Podolsky mass μ. Loop integrals in self-energy calculations require $\Lambda$ to be set by physical parameters—such as inverse lattice spacing in condensed matter scenarios. The screening effect dictated by μ restricts effective coupling: $m^2 \leq k^2 < \mu^2$ where loss of predictivity occurs at $k^2 \sim \mu^2$, owing to singularities in the running coupling and breakdown of perturbative expansion. In practice, the regime of validity for PGQED is thus: $$\alpha < \alpha_c(\mu), \quad N > N_c(\mu), \quad k^2 < \mu^2$$

6. Phenomenological Applications: Graphene and Strongly Correlated Systems

PGQED’s planar geometry and screened interaction model the physics of two-dimensional Dirac materials, such as graphene. Here, electrons behave as massless Dirac fermions in 2D, while the gauge field remains three-dimensional, reflecting unscreened long-range Coulomb interactions:

• The static potential reproduces the nonlocal screening features found in experiment.
• Estimates for $\Lambda \sim 1$ eV (inverse lattice scale) and $\alpha \simeq 2.2$ for suspended graphene allow parameter matching.
• Measured gaps ($\sim 0.1$ meV) correspond to Podolsky screening masses $\mu \sim 0.2$ meV, suggesting PGQED captures essential nonperturbative electron–electron correlations in realistic materials.

7. Broader Implications and Future Directions

PGQED demonstrates that dimensional reduction of higher-derivative gauge theories produces effective field theories with improved ultraviolet behavior, nonlocal screening, and physically relevant mass gaps. The framework supports further exploration:

• Analysis with finite Fermi velocity to model realistic graphene is necessary, as $v_F \ll c$ violates Lorentz invariance.
• Inclusion of external fields (e.g., magnetic) for quantum Hall regimes.
• Quantum simulation in optical lattice setups where emergent gauge fields with nonlocal kinetic terms might be engineered.

PGQED provides explicit analytic and robust numerical results for critical phenomena in planar Dirac systems, allowing clear criteria for phase transitions and band gap formation due to many-body effects. It bridges traditional field-theoretical renormalization techniques with the phenomenology of two-dimensional quantum materials, guided by concrete experimental bounds and consistent UV regularization.