Dynamical mass generation and critical behavior in pseudo-Proca quantum electrodynamics

Helio G. Barroso; Leandro O. Nascimento; Van S\'ergio Alves

REVIEW 5 major objections 5 minor 50 references

Dynamical mass generation and critical behavior in pseudo-Proca quantum electrodynamics

T0 review · 5 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Adding a Proca mass to planar QED makes chiral symmetry breaking harder, because the mass acts as a Yukawa-screening knob that suppresses dynamical fermion mass generation.

desk verdict Plausible physics undone by an uncontrolled approximation and a sign error in the printed equations; the qualitative picture may survive, but the quantitative formulas are not reliable. read the letter →

arxiv 2510.19134 v3 pith:YBYZKHRZ submitted 2025-10-21 hep-th

Helio G. Barroso , Van Sérgio Alves , Leandro O. Nascimento This is my paper · ORCID

classification hep-th MSC 81T1381T1681V10

keywords pseudo-ProcaQEDdynamicalmassgenerationchiralsymmetrybreakingSchwinger-DysonequationscriticalcouplingYukawascreeningfermionflavornumberanisotropicFermivelocity

topics Yang-Mills Mass Gap

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that in pseudo-Proca quantum electrodynamics, a (2+1)-dimensional theory of planar fermions interacting via a massive gauge mode inherited from (3+1)D Proca electrodynamics, the gauge-field mass m acts as a tunable screening scale that controls the chiral phase transition. Using Schwinger–Dyson equations in rainbow-quenched and unquenched truncations, the authors derive closed-form critical couplings: α_c(m,Λ) = (π/8)(1 + m²/Λ²)^{3/2} in the quenched case, and a critical flavor number N_c(m,Λ,g) that shrinks with m. Both trends mean that increasing m suppresses dynamical mass generation: a stronger coupling or fewer fermion flavors is required to break chiral symmetry. If correct, this yields a simple physical picture—Yukawa screening shortens the interaction range and thereby weakens the pairing that drives mass generation—and connects to Proca metamaterials where an effective photon mass is tunable in the same m/Λ window.

What carries the argument

The central object is the nonlocal gauge-field propagator of pseudo-Proca QED, ∆_{µν}(k) = δ_{µν}/(2√(k²+m²)) in Landau gauge, obtained by dimensional reduction from the (3+1)D Proca–Stueckelberg theory. The mass parameter m acts as a screening scale: the static potential becomes the Yukawa potential e^{-mr}/(4πr), and in the Schwinger–Dyson gap equation m enters through factors (p²+m²)^{1/2} in the kernel. The load-bearing step in the analytical derivation is the replacement p²(1 + m²/p²)^{3/2} ≈ p²(1 + m²/Λ²)^{3/2}, justified by the claim that the integrand peaks near the ultraviolet cutoff Λ; this replacement linearizes the differential equation and yields closed-form expressions for α_c,

What would settle it

Numerically solve the full integral gap equation (Eq. 23 or the unquenched analog, Eq. 43) without the p ≈ Λ linearization, over a range of m/Λ from 0 to 1, and compare the extracted α_c(m) and N_c(m) against Equations (35) and (51); the central claim fails if the exact curves do not increase (for α_c) or decrease (for N_c) monotonically with m, or if the deviation is large in the regime where the approximation is supposed to hold (m ≪ Λ).

Watch

Extended reading notes

Core claim

In pseudo-Proca QED, the massive gauge mode acts as a screening scale m that enters the gap equation through the effective propagator 1/(2√(k²+m²)). The paper's central result is that this mass suppresses dynamical chiral symmetry breaking: in the rainbow-quenched approximation the critical fine-structure constant increases as α_c = (π/8)(1 + m²/Λ²)^{3/2}, and in the 1/N unquenched approximation the critical flavor number decreases as N_c = 2g/[π² f1(m,Λ,g)]. The mass converts the long-range Coulomb potential into a short-range Yukawa form e^{-mr}/(4πr), removing the low-momentum support that drives fermion binding. The same suppression appears in the anisotropic case with Fermi velocity v_F

Load-bearing premise

The entire m-dependence of the critical couplings comes from replacing the momentum-dependent factor (1 + m²/p²)^{3/2} in the gap equation by its value at the ultraviolet cutoff, (1 + m²/Λ²)^{3/2}, on the grounds that the integrand peaks near p ∼ Λ; if the relevant momentum range that fixes the critical solution is not in that region, the predicted sign and size of the m-dependence would change.

Editorial extensions

If this is right

Dynamical mass generation becomes harder as m increases: α_c grows monotonically with m/Λ, so the chiral phase transition requires stronger coupling.
The critical number of flavors N_c decreases with m, meaning fewer fermion flavors are allowed before chiral symmetry is restored in the unquenched theory.
In anisotropic Dirac materials, raising the Fermi velocity v_F relative to c raises the critical coupling, so mass generation is suppressed; for realistic v_F ∼ c/300–c/100, α*_c exceeds α_c.
For m → 0, all critical parameters reduce to the known PQED and QED3 results, confirming that PPQED continuously interpolates between Coulomb and Yukawa regimes.
The screening-driven suppression is qualitatively robust to vertex corrections: a Ball–Chiu vertex changes quantitative values but preserves the trend.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the mechanism is interaction-range driven, one would expect analogous suppression in any gap-equation model with a screened kernel, for example four-fermion theories with a mass scale, though the precise exponents would differ.
A natural extension is to study finite-temperature or finite-density PPQED, where thermal fluctuations or chemical potential will compete with Yukawa screening and may shift the critical surface; the authors list these as future directions.
The connection to Proca metamaterials suggests a classical analogue: in a patterned medium with tunable effective photon mass, some threshold response (such as transmission or absorption) might exhibit a mass-controlled shift, though a quantitative interface model is needed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me give you the short version first: this paper has a sensible physical idea and a competent setup, but the printed derivation undercuts the headline claim. The central m-dependence of α_c and N_c rests on an uncontrolled ultraviolet approximation, and one of the key equations has the wrong sign in the m-dependent coefficient. I would not take the quantitative formulas at face value.

What's good: the authors apply the standard rainbow and 1/N Schwinger–Dyson formalism to pseudo-Proca QED, and the m→0 limits correctly reduce to the known PQED and QED3 results. The qualitative picture—that a Proca mass screens the interaction and suppresses dynamical mass generation—is physically reasonable. The corrected static potential in Appendix B is a useful side result.

The soft spots are serious. In Section IV, the approximation in Eq. (32) replaces the momentum-dependent coefficient (1+m^2/p^2)^{3/2} by its value at the cutoff. That is not controlled: the critical coupling in a linearized gap equation is set by the indicial exponent and the infrared boundary condition, not by the value of the coefficient at one momentum. The printed Eq. (33) is worse: starting from Eq. (31), the coefficient (1+m^2/Λ^2)^{3/2} should appear in the denominator of the coupling term, not the numerator. As printed, Eq. (33) gives α_c decreasing with m, the opposite of the claimed Eq. (35). The numerical appendix cannot rescue this because it only tests m/Λ = 10^{-5} and 10^{-2}, where Eq. (35) differs from π/8 by a fraction of a percent; the claimed 10–20% effect is never tested.

There are smaller but real issues. The abstract mentions a Ball–Chiu vertex construction that never appears in the body. In Section V, one sentence says N_c increases with m while the formula and Fig. 3 show it decreases. The anisotropic section is internally inconsistent: Eq. (63) has (1+v_F^2/c^2) in the numerator, but the text after it says increasing v_F raises α_c, and the summary says the scaling is (1+v_F^2/c^2)^{-1}; the correct derivation from Eq. (61) actually gives the inverse, so Eq. (63) is simply wrong.

This is not a paper to desk-reject, but it is not near acceptance. A serious referee could help the authors fix the equations, replace the uncontrolled approximation with a controlled one (or at least an error estimate), and run the numerics at m/Λ values where the effect is actually visible. The physical direction is plausible; the paper as printed does not support its quantitative claims.

Referee Report

5 major / 5 minor

Summary. The manuscript studies pseudo-Proca QED (PPQED) in (2+1) dimensions obtained by dimensional reduction from a (3+1)-dimensional Proca-Stueckelberg theory. Using Schwinger-Dyson equations in rainbow-quenched and rainbow-unquenched truncations, it derives analytical estimates for the critical fine-structure constant and the critical fermion flavor number: α_c(m,Λ)=π/8(1+m²/Λ²)^{3/2}, N_c(m,Λ,g)=2g/[π² f1(m,Λ,g)], together with an anisotropic static extension giving α_c^*=(1/2)(1+m²/Λ²)^{3/2}(1+v_F²/c²). The central physical claim is that the Proca screening scale m suppresses dynamical chiral symmetry breaking, with α_c increasing and N_c decreasing with m/Λ, and that finite Fermi velocity acts in the same direction. The paper also relates the results to Proca metamaterials and reports numerical solutions of the integral equations in appendices.

Significance. If the derivation were sound, the paper would establish a tunable screening-mass control of the chiral phase transition in a dimensionally reduced gauge theory, with explicit analytic formulas, no fitted parameters, and the expected Miransky-type scaling near criticality. The paper's strengths include a clean derivation of the nonlocal PPQED propagator and static Yukawa potential, correct limiting recovery of PQED/QED3 results when m→0, and numerical solutions for Σ(p), A(p), and B(p) using the repeated trapezoidal method. However, the central quantitative mass dependence rests on an uncontrolled replacement of a momentum-dependent coefficient by its value at the cutoff, and the printed derivation contains internal inconsistencies (notably the coefficient in Eq. (33) and the abstract's anisotropic claim). The advertised Ball-Chiu vertex analysis is absent from the body. The qualitative screening picture is plausible, but the quantitative formulas are not presently supported.

major comments (5)

[Abstract vs. body (Sections II-VII, Appendices A-B)] The abstract twice states that the robustness of the results was assessed by incorporating a Ball-Chiu vertex construction. No Ball-Chiu vertex analysis appears anywhere in the main text or in the appendices. This is advertised as a substantive contribution, and its absence is a load-bearing discrepancy. The authors must either include the missing analysis or revise the abstract to remove the claim.
[Section IV, Eq. (33) vs. Eq. (35)] Eq. (33) is inconsistent with the derivation preceding it. Substituting p²(1+m²/p²)^{3/2} ≈ p²(1+m²/Λ²)^{3/2} into Eq. (31) gives p²Σ''+2pΣ' + (2α/π)(1+m²/Λ²)^{-3/2} Σ = 0, not the equation with the positive power printed in Eq. (33). Solving Eq. (33) literally yields α_c = π/8 (1+m²/Λ²)^{-3/2}, which decreases with m and contradicts Eq. (35). The numerical checks in Appendix A use m/Λ = 10^{-5} and 10^{-2}, for which the difference between the two expressions is below 0.02%, so the numerics cannot resolve the contradiction or validate the 10-20% effect shown in Fig. 3. The coefficient must be corrected and the numerical determination of α_c must be reported at the m/Λ values for which the effect is claimed.
[Section IV, Eq. (32); also Section V, Eqs. (44)-(48)] The replacement of the momentum-dependent coefficient p²(1+m²/p²)^{3/2} by p²(1+m²/Λ²)^{3/2} is the only source of the claimed m dependence, and it is not controlled. The exact coefficient is singular at p→0, the IR indicial behavior of Eq. (31) differs from that of the constant-coefficient ODE, and the critical coupling is fixed by matching the solution to the IR boundary condition. The statement that 'the integrand peaks near Λ' does not justify evaluating the coefficient at p=Λ, and no error estimate is given. The same issue appears in the unquenched case, where f1(m,p,g) is replaced by f1(m,Λ,g) in Eqs. (44)-(48). The authors should either provide a systematic expansion in m/Λ with controlled errors, or solve the full momentum-dependent ODE/integral equation numerically and compare the resulting α_c(m,Λ) and N_c(m,Λ,g) with the proposed formulas.
[Section V, text after Eq. (51)] The paragraph after Eq. (51) states that for m/Λ ∼ 10^{-2}–10^{-1} the critical flavor number N_c increases by approximately 10–20% relative to the massless limit. This contradicts Eq. (51) and Fig. 3, both of which show N_c decreasing with m/Λ; moreover, for m/Λ=0.1 the decrease predicted by Eq. (51) is sub-percent, not 10-20%. This internal inconsistency affects the paper's central summary and the abstract's physical claim. The authors must correct the sentence and state which quantity changes in which direction and by how much.
[Abstract and Section VI, Eq. (63)] The abstract states that the anisotropic critical coupling decreases as v_F/c increases, while Eq. (63) gives α_c^* = (1/2)(1+m²/Λ²)^{3/2}(1+v_F²/c²), which increases with v_F/c, and the text around Eq. (63) explicitly says increasing v_F raises the critical coupling. In addition, Eq. (63) does not reduce to the isotropic result Eq. (35) when v_F → c; it gives 1 rather than π/8, because the anisotropic calculation uses a different static approximation. This limits the direct comparison α_c^* > α_c in Section VI, and the abstract must be corrected.

minor comments (5)

[Eq. (32)] The displayed approximation is missing the exponent 3/2: it reads p²(1+m²/p²) ≈ p²(1+m²/Λ²), whereas the text and the following equation require p²(1+m²/p²)^{3/2} ≈ p²(1+m²/Λ²)^{3/2}.
[Appendix A, figure captions] The notation 'uΛ' is used in the captions of Figs. 4-8 without definition. Please define the units of the dimensionless momentum/mass variables.
[Sections II and VI] The paper sets c=1 in Section II but restores c explicitly in the anisotropic section. The conventions should be stated more carefully, especially in Eq. (54) and (55), so that the v_F/c factors are unambiguous.
[Reference [27] and Fig. 9 caption] Reference [27] contains a garbled title ('Greenˆ a€™s functions'); the caption of Fig. 9 has 'Fig.. 9'. These should be corrected in production.
[Eq. (38)] The prefactor involving (Λ²−m²) diverges at m=Λ. The text calls this an artifact of the m≪Λ approximation, but the restriction m≪Λ should be stated explicitly wherever Eq. (38) and similar formulas are used.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the mass dependence of alpha_c and N_c follows from a declared linearization of the gap equation; self-citations are used only as m=0 consistency checks.

full rationale

Walking the derivation chain from the PPQED action (Sec. II), through the rainbow-quenched gap equation (Eqs. 23-28), the linearized critical-coupling extraction (Eqs. 31-35), the unquenched 1/N treatment (Eqs. 43-51), and the anisotropic extension (Eqs. 58-63), I find no step in which an output is equivalent by construction to an input. The m-dependence is generated by the explicitly stated replacement in Eq. (32), p^2(1+m^2/p^2) -> p^2(1+m^2/Lambda^2); this is a declared approximation, not a hidden fit, and the critical coupling then follows from a standard indicial-equation solution. No fitted parameter is renamed as a prediction: the only fits in Appendix A are auxiliary constants A1, A2, C1, C2 used to overlay analytic forms onto numerical curves, not the critical parameters themselves. The citations to the authors' earlier PQED work enter as m=0 consistency checks and as model-building background; the mass-dependent claim does not rest on a self-citation or on an imported uniqueness theorem. I note that Eq. (33) as printed places (1+m^2/Lambda^2)^{3/2} in the numerator, whereas consistency with Eq. (31) after the same replacement would put it in the denominator; this is an internal algebraic self-consistency problem that would invert the printed sign of d(alpha_c)/dm, but an algebraic error is not circularity under the specified rubric. The central derivation is therefore self-contained rather than circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard rainbow/1/N truncation plus one ad hoc linearization step (Eq. 32) that generates the full m-dependence. No new entities are postulated. The most fragile input is the uncontrolled replacement of p by Λ in the mass term, compounded by an algebraic sign inconsistency in the printed Eq. (33).

assumptions (5)

domain assumption The pseudo-Proca propagator in Landau gauge is Δ(0)_{μν}(k) = (1/(2√(k²+m²)))(δ_{μν} − k_μk_ν/k²) and captures the dimensionally reduced Proca theory.
Eq. (17), taken from [18]. This is the starting point for all gap equations and for the claimed Yukawa screening picture.
domain assumption The rainbow truncation with A(p)=1 (and B(p)=1 in the anisotropic case) is quantitatively reliable for the qualitative critical behavior.
Sections III and VI; the authors state a fully self-consistent solution lies beyond scope and note deviations below 1% only in the symmetric phase for a few parameter values.
ad hoc to paper The replacement p²(1+m²/p²)^{3/2} ≈ p²(1+m²/Λ²)^{3/2} is valid because the integrand peaks near the UV cutoff.
Eq. (32); this is the load-bearing linearization used to obtain closed-form α_c and N_c. No error bound is given, and the printed Eq. (33) has a coefficient inconsistent with it.
domain assumption The 1/N leading-order polarization Π(p²) = −(g/8)√p² is unchanged by the pseudo-Proca mass.
Section V, Eq. (41). The authors assert this is identical at leading order in PPQED; if m-dependent corrections enter at this order, the N_c formula changes.
domain assumption Gauge dependence of intermediate critical quantities is mild and the physical picture is robust across gauges.
Section II relies on QED3 studies [31]; this is not demonstrated for PPQED.

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Cite this review

Pith. "Pith review of Dynamical mass generation and critical behavior in pseudo-Proca quantum electrodynamics." pith.science (2026). https://pith.science/paper/YBYZKHRZ

@misc{pith2026251019134,
  author       = {Pith},
  title        = {Pith review of: Dynamical mass generation and critical behavior in pseudo-Proca quantum electrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YBYZKHRZ}},
  note         = {Machine review of arXiv:2510.19134}
}

abstract

We investigate dynamical mass generation in pseudo-Proca quantum electrodynamics (PPQED) by means of Schwinger--Dyson equations in rainbow-quenched and unquenched truncations. The pseudo-Proca screening scale $m$, together with the fine-structure constant $\alpha$, the flavor number $N$, and the ultraviolet cutoff $\Lambda$, control the critical thresholds for chiral symmetry breaking in the reduced $(2+1)$D theory. Within these truncation schemes, we obtain analytical estimates for both the critical coupling $\alpha_c(m,\Lambda)$ and the critical number of fermion flavors $N_c(m,\Lambda,g)$, and show that increasing $m$ suppresses dynamical mass generation through Yukawa screening. We further assess the robustness of this physical picture by incorporating a Ball--Chiu vertex construction, which modifies the quantitative values of the critical parameters while preserving their qualitative dependence on the screening scale $m$. In addition, within the static approximation, the anisotropic extension indicates that the critical coupling decreases as the ratio $v_F/c$ increases. Taken together, our results suggest that the scale $m$ plays an important role in modulating criticality in PPQED. Furthermore, the Ball-Chiu vertex construction shows that vertex corrections modify the quantitative values of the critical parameters while preserving the screening-driven suppression of dynamical mass generation in $(2+1)$D.

Figures

Figures reproduced from arXiv: 2510.19134 by the authors.

**Figure 2.** The gauge field SD equation. Filled dots indicate [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Dependence of the normalized critical coupling [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Numerical solution of the mass function Σ( [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 6.** Figure 6: Numerical solutions of Eq. (58) are presented for two mass values and two Fermi velocities (vF = c/300 and vF = c/250). For these mass values, α ∗ c ≈ 0.5. Each plot includes curves for two coupling constants: α = 0.23 (black) and α = 1.5 (blue and red), facilitating t…

**Figure 7.** Figure 7: Numerical solutions of Eq. (A3) for different mass values (m = 1 × 10−5 uΛ and m = 1 × 10−2 uΛ). The figure displays the curves for α = 0.1. As can be obtained from Eq. (57), the Fermi velocity renormalization function is B(p)|Σ=0 = 1 + πα 1 + v 2 F c 2 Z Λ 0 dk (2…

**Figure 9.** Figure 9: Numerical solution for the corrected static potential [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 8.** Figure 8: Numerical solutions of the Fermi velocity renormal [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

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Works this paper leans on

50 extracted references · 2 canonical work pages

[1]

Mikki, Proca metamaterials, massive electromag- netism, and spatial dispersion, Annalen der Physik533, 2000625 (2021)

S. Mikki, Proca metamaterials, massive electromag- netism, and spatial dispersion, Annalen der Physik533, 2000625 (2021)

2021
[2]

P. A. M. Dirac, Quantum theory of emission and ab- sorption of radiation, Proc. Roy. Soc. Lond. A114, 243 (1927); The quantum theory of the electron, Proc. Roy. Soc. Lond. A117, 610 (1928)

1927
[3]

Tomonaga, On a relativistically invariant formulation of the quantum theory of wave fields, Prog

S. Tomonaga, On a relativistically invariant formulation of the quantum theory of wave fields, Prog. Theor. Phys. 1, 27 (1946)

1946
[4]

Schwinger, On quantum-electrodynamics and the mag- netic moment of the electron, Phys

J. Schwinger, On quantum-electrodynamics and the mag- netic moment of the electron, Phys. Rev.73, 416 (1948)

1948
[5]

R. P. Feynman, Space-time approach to quantum elec- trodynamics, Phys. Rev.76, 769 (1949)

1949
[6]

F. J. Dyson, The radiation theories of tomonaga, schwinger, and feynman, Phys. Rev.75, 486 (1949)

1949
[7]

Marino, Quantum electrodynamics of particles on a plane and the chern-simons theory, Nuclear Physics B 408, 551 (1993)

E. Marino, Quantum electrodynamics of particles on a plane and the chern-simons theory, Nuclear Physics B 408, 551 (1993)

1993
[8]

E. V. Gorbar, V. P. Gusynin, and V. A. Miransky, Dy- namical chiral symmetry breaking on a brane in reduced qed, Phys. Rev. D64, 105028 (2001)

2001

Show all 50 references

[9]

Kovner and B

A. Kovner and B. Rosenstein, Kosterlitz-thouless mech- anism of two-dimensional superconductivity, Phys. Rev. B42, 4748 (1990)

1990
[10]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys.81, 109 (2009)

2009
[11]

E. V. Gorbar, V. P. Gusynin, and M. R. Parymuda, Re- duced qed with few planes and fermion gap generation, Entropy25, 10.3390/e25091317 (2023)

2023 doi
[12]

E. H. Hwang and S. Das Sarma, Dielectric function, screening, and plasmons in two-dimensional graphene, Phys. Rev. B75, 205418 (2007); V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, Electron-electron interactions in graphene: Current sta- tus and persp...

2007
[13]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric field effect in atomically thin carbon films, Science306, 666 (2004); N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Electronic properties of disorder...

2004
[14]

E. C. Marino, L. O. Nascimento, V. S. Alves, and C. M. Smith, Interaction induced quantum valley hall effect in graphene, Phys. Rev. X5, 011040 (2015)

2015
[15]

P. W. Higgs, Broken symmetries and the masses of gauge bosons, Phys. Rev. Lett.13, 508 (1964); S. Weinberg, A model of leptons, Phys. Rev. Lett.19, 1264 (1967); J. F. Gunion, H. E. Haber, G. L. Kane, and S. Dawson,The Higgs Hunter’s Guide, 1st ed. (CRC Press, 1990)

1964
[16]

E. C. G. Stueckelberg, Interaction energy in electrody- namics and in the field theory of nuclear forces, Helv. Phys. Acta11, 225 (1938)

1938
[17]

Ruegg and M

H. Ruegg and M. Ruiz-Altaba, The stueckelberg field, In- ternational Journal of Modern Physics A19, 3265 (2004)

2004
[18]

V. S. Alves, T. Macr ` ı, G. C. Magalh˜ aes, E. C. Marino, and L. O. Nascimento, Two-dimensional yukawa inter- actions from nonlocal proca quantum electrodynamics, Phys. Rev. D97, 096003 (2018)

2018
[19]

R. F. Ozela, V. S. Alves, L. O. Nascimento, E. C. Marino, C. M. Smith, R. O. Ramos, and J. F. M. Neto, Effective mass in mixed-dimensionality pseudo-proca systems,g- factor corrections, and the radiative effects to the inter- action potential, Phys. Rev. D108, 056017 (2023)

2023
[20]

Johnson, M

K. Johnson, M. Baker, and R. Willey, Self-energy of the electron, Phys. Rev.136, B1111 (1964); T. Maskawa and H. Nakajima, Spontaneous breaking of chiral sym- metry in a vector-gluon model. ii, Progress of Theoret- ical Physics54, 860 (1975); R. Fukuda and T. Kugo, Schwinger-d...

1964
[21]

Appelquist and R

T. Appelquist and R. D. Pisarski, High-temperature yang-mills theories and three-dimensional quantum chro- modynamics, Phys. Rev. D23, 2305 (1981); T. Ap- pelquist, D. Nash, and L. C. R. Wijewardhana, Critical behavior in (2+1)-dimensional qed, Phys. Rev. Lett.60, 2575 (1988)

1981
[22]

Nash, Higher-order corrections in (2+1)-dimensional qed, Phys

D. Nash, Higher-order corrections in (2+1)-dimensional qed, Phys. Rev. Lett.62, 3024 (1989); P. Maris, Confine- ment and complex singularities in three-dimensional qed, Phys. Rev. D52, 6087 (1995); E. Dagotto, J. B. Kogut, and A. Koci´ c, Computer simulation of chiral-symmetry...

1989
[23]

V. S. Alves, W. S. Elias, L. O. Nascimento, V. Juriˇ ci´ c, and F. Pe˜ na, Chiral symmetry breaking in the pseudo- quantum electrodynamics, Phys. Rev. D87, 125002 (2013)

2013
[24]

V. S. Alves, R. O. C. Junior, E. C. Marino, and 14 L. O. Nascimento, Dynamical mass generation in pseudo- quantum electrodynamics with four-fermion interactions, Phys. Rev. D96, 034005 (2017)

2017
[25]

Fern´ andez, R

L. Fern´ andez, R. O. Corrˆ ea, V. S. Alves, L. O. Nasci- mento, and F. Pe˜ na, Dynamical mass generation in pseu- doquantum electrodynamics with gross-neveu interaction at finite temperature, Phys. Rev. D103, 025018 (2021)

2021
[26]

F. J. Dyson, Thesmatrix in quantum electrodynamics, Phys. Rev.75, 1736 (1949)

1949
[27]

Schwinger, On the greenˆ a€™s functions of quantized fields

J. Schwinger, On the greenˆ a€™s functions of quantized fields. ii, Proceedings of the Na- tional Academy of Sciences37, 455 (1951), https://www.pnas.org/doi/pdf/10.1073/pnas.37.7.455

1951 doi
[28]

Proca, Sur la theorie ondulatoire des electrons positifs et negatifs, J

A. Proca, Sur la theorie ondulatoire des electrons positifs et negatifs, J. Phys. Radium7, 347 (1936)

1936
[29]

Yukawa, On the interaction of elementary particles

H. Yukawa, On the interaction of elementary particles. i, Nippon Sugaku-Buturigakkwai Kizi Dai 3 Ki17, 48 (1935)

1935
[30]

L. O. Nascimento, C. A. P. C. Junior, and J. R. San- tos, The effective potential of scalar pseudo-quantum electrodynamics in (2 + 1)d, Condensed Matter9, 10.3390/condmat9020025 (2024)

2024 doi
[31]

T. W. Appelquist, M. Bowick, D. Karabali, and L. Wijewardhana, Spontaneous chiral-symmetry break- ing in three-dimensional qed, Physical Review D33, 3704 (1986); P. Maris and C. D. Roberts, Dyson- schwinger equations: A tool for hadron physics, Inter- national Journal of Moder...

1986 doi
[32]

C. D. Roberts and A. G. Williams, Dyson-schwinger equations and their application to hadronic physics, Progress in Particle and Nuclear Physics33, 477 (1994)

1994
[33]

Alkofer and L

R. Alkofer and L. von Smekal, The infrared behaviour of qcd green’s functions: Confinement, dynamical symme- try breaking, and hadrons as relativistic bound states, Physics Reports353, 281 (2001)

2001
[34]

In (2+1)D Euclidean space and in the 4×4 repre- sentation: Tr(γ µγν ) =−4δ µν , Tr(γ µγν γα) = 0, and Tr(γµγν γαγβ) = 4(δµν δαβ −δ µαδνβ +δ µβδνα )
[35]

Bashir and A

A. Bashir and A. Raya, Dynamical fermion masses and constraints of gauge invariance in quenched qed3, Nuclear Physics B709, 307 (2005)

2005
[36]

Atkinson and P

D. Atkinson and P. W. Johnson, Bifurcation of the quark self-energy: Infrared and ultraviolet cutoffs, Phys. Rev. D35, 1943 (1987)

1943
[37]

P. I. Fomin, V. P. Gusynin, V. A. Miransky, and Y. A. Sitenko, Dynamical Symmetry Breaking and Particle Mass Generation in Gauge Field Theories, Riv. Nuovo Cim.6N5, 1 (1983)

1983
[38]

V. A. Miransky, Dynamics of spontaneous chiral symme- try breaking and the continuum limit in quantum elec- trodynamics, Il Nuovo Cimento A (1965-1970)90, 149 (1985); V. Miransky, On dynamical chiral symmetry breaking, Physics Letters B165, 401 (1985)

1965
[39]

R. D. Pisarski, Chiral-symmetry breaking in three- dimensional electrodynamics, Phys. Rev. D29, 2423 (1984)

1984
[40]

V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Dy- namical flavor symmetry breaking by a magnetic field in 2+1 dimensions, Phys. Rev. D52, 4718 (1995)

1995
[41]

Isobe and N

H. Isobe and N. Nagaosa, Theory of a quantum crit- ical phenomenon in a topological insulator: (3+1)- dimensional quantum electrodynamics in solids, Phys. Rev. B86, 165127 (2012)

2012
[42]

Isobe and N

H. Isobe and N. Nagaosa, Renormalization group study of electromagnetic interaction in multi-dirac-node systems, Phys. Rev. B87, 205138 (2013)

2013
[43]

Isobe and N

H. Isobe and N. Nagaosa, Coulomb interaction effect in weyl fermions with tilted energy dispersion in two dimen- sions, Phys. Rev. Lett.116, 116803 (2016)

2016
[44]

Popovici, C

C. Popovici, C. S. Fischer, and L. von Smekal, Fermi velocity renormalization and dynamical gap generation in graphene, Phys. Rev. B88, 205429 (2013)

2013
[45]

Dorey and N

N. Dorey and N. Mavromatos, Qed3 and two-dimensional superconductivity without parity violation, Nuclear Physics B386, 614 (1992)

1992
[46]

Gonz´ alez, F

J. Gonz´ alez, F. Guinea, and M. Vozmediano, Non-fermi liquid behavior of electrons in the half-filled honey- comb lattice (a renormalization group approach), Nuclear Physics B424, 595 (1994)

1994
[47]

I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products, 8th ed., edited by D. Zwillinger and V. Moll (Academic Press, Amsterdam, 2014)

2014
[48]

M. I. Katsnelson, Graphene: carbon in two dimensions, Materials Today10, 20 (2007)

2007
[49]

Wang, T.-Y

Z.-Q. Wang, T.-Y. L¨ u, H.-Q. Wang, Y. P. Feng, and J.- C. Zheng, Review of borophene and its potential appli- cations, Frontiers of Physics14, 33403 (2019)

2019
[50]

W. H. Press and M. Metcalf,Numerical recipes in Fortran 90 : the art of parallel scientific computing : volume 2 of fortran numerical recipes, 2nd ed. (Cambridge University Press, 1999)

1999

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Mikki, Proca metamaterials, massive electromag- netism, and spatial dispersion, Annalen der Physik533, 2000625 (2021)

S. Mikki, Proca metamaterials, massive electromag- netism, and spatial dispersion, Annalen der Physik533, 2000625 (2021)

2021

[2] [2]

P. A. M. Dirac, Quantum theory of emission and ab- sorption of radiation, Proc. Roy. Soc. Lond. A114, 243 (1927); The quantum theory of the electron, Proc. Roy. Soc. Lond. A117, 610 (1928)

1927

[3] [3]

Tomonaga, On a relativistically invariant formulation of the quantum theory of wave fields, Prog

S. Tomonaga, On a relativistically invariant formulation of the quantum theory of wave fields, Prog. Theor. Phys. 1, 27 (1946)

1946

[4] [4]

Schwinger, On quantum-electrodynamics and the mag- netic moment of the electron, Phys

J. Schwinger, On quantum-electrodynamics and the mag- netic moment of the electron, Phys. Rev.73, 416 (1948)

1948

[5] [5]

R. P. Feynman, Space-time approach to quantum elec- trodynamics, Phys. Rev.76, 769 (1949)

1949

[6] [6]

F. J. Dyson, The radiation theories of tomonaga, schwinger, and feynman, Phys. Rev.75, 486 (1949)

1949

[7] [7]

Marino, Quantum electrodynamics of particles on a plane and the chern-simons theory, Nuclear Physics B 408, 551 (1993)

E. Marino, Quantum electrodynamics of particles on a plane and the chern-simons theory, Nuclear Physics B 408, 551 (1993)

1993

[8] [8]

E. V. Gorbar, V. P. Gusynin, and V. A. Miransky, Dy- namical chiral symmetry breaking on a brane in reduced qed, Phys. Rev. D64, 105028 (2001)

2001