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REVIEW 4 major objections 4 minor 26 references

Electric field fluctuations and renormalization group flows in a self-interacting scalar field theory

T0 review · 4 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Stochastic electric-field fluctuations turn scalar self-interactions complex and alter their renormalization flow — weakly, toward Landau poles; very strongly, toward a pole-free, quasi-free regime.

desk verdict Original replica-trick mechanism for scalar fields in noisy electric backgrounds, but the one-loop renormalization conditions and counterterms are inconsistent, so the beta functions are not yet supported. read the letter →

arxiv 2607.16932 v1 pith:ZM6UIXTJ submitted 2026-07-18 hep-th hep-ph

Melanie Martínez , Enrique Muñoz , Marcelo Loewe This is my paper · ORCID
classification hep-thhep-ph
keywords complexscalarfieldtheorystochasticbackgroundelectricreplicatrickSchwingerproper-timepropagatorrenormalizationgroupbetafunctionscurrent-currentinteractionasymptoticfreedom
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

This paper studies a complex λφ⁴ scalar field (a toy model for charged pions) placed in a background electric field whose fluctuations are classical white noise. After averaging over the noise with the replica trick, the theory acquires a new effective interaction between charged currents, with a strength set by the noise auto-correlation Δ. The paper then computes one-loop self-energies, vertices, and renormalization-group equations in two regimes: very weak electric field (|qE|/m² ≪ 1) and very strong field (|qE|/m² ≫ 1). The central result is that the noise makes both couplings complex and changes their evolution: in the weak-field regime the real parts grow and hit Landau poles, with the noise damping the λ-coupling, while in the strong-field regime the poles disappear and the theory behaves as if asymptotically free, so particles behave as free at high energy. A sympathetic reader would care because heavy-ion collisions produce strong, short-lived electric fields, and if this toy-model behavior survives more realistic noise, fluctuating backgrounds could materially affect effective descriptions of pion matter.

What carries the argument

The argument runs on two pieces. First, the replica trick: writing the generating functional as an n-fold replicated theory and Gaussian-integrating over the white-noise fluctuations ⟨δAμδAν⟩ = Δ δμν δ⁴(x−y) converts the noise into a local, bilinear current-current vertex -i q² Δ Σ_{a,b} j_{a,μ} j_b^μ. Second, the Schwinger proper-time propagator for a uniform electric field, taken in the two limits |qE|/m² ≪ 1 (weak-field expansion, giving the free propagator plus (qE)² corrections) and |qE|/m² ≫ 1 (Landau-level representation, dominated by the l=0 mode with effective mass m_E² = qE + m²). These two ingredients feed the one-loop self-energy and vertex integrals that produce the counterterms

What would settle it

Take the same complex λφ⁴ model with a noise correlator of finite width τ (e.g. ⟨δAμδAν⟩ = Δ δμν (τ²/4π) e^{-|x−y|²/4τ²}) and recompute the one-loop self-energy and vertex. If the effective interaction is non-local, Eqs. (78)-(79) and (85)-(86) cannot hold as written; the β-functions will acquire momentum dependence and the pole-free strong-field flow may disappear. Alternatively, compute the two-loop counterterms: if the one-loop counterterms in Sec. V fail to cancel the two-loop divergences, the replica-trick-before-renormalization order is not justified.

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Extended reading notes

Core claim

The paper's central claim is that after Gaussian averaging over electric-field noise, the effective Lagrangian acquires a current-current interaction -i q² Δ Σ_{a,b} j_{a,μ} j_b^μ whose dimensionless strength Δ~ = q² m² Δ runs with energy. In the weak-field limit the one-loop beta functions are β_{Δ~} = (2/π²) Δ~² (1 - i/4) and β_λ = 3/(16π²) λ² - i/(2π²) λ Δ~; in the strong-field limit they are β_{Δ~} = -i (qE)/(π² m²) Δ~² and β_λ = -i (qE)/(π² m²) λ Δ~. The paper argues that these flows imply Landau poles in weak fields—with the noise acting as a damping factor for the self-coupling—and, in very strong fields, a pole-free flow interpreted as asymptotic freedom, so that the dressed spectral

Load-bearing premise

The derivation breaks if the noise is not white: the assumption ⟨δAμ(x)δAν(y)⟩ = Δ δμν δ⁴(x−y) is what makes the replica-averaged interaction local, and with a colored correlator the current-current vertex becomes non-local and the computed beta functions no longer follow.

Editorial extensions

If this is right

  • If correct, the noise changes the UV behavior of the scalar theory: weak-field flows are cut off by Landau poles at t < π²/(2Δ~₀) for the noise coupling, so the scale of physics is bounded.
  • In the strong-field regime the couplings run without Landau poles, so a very strong background electric field would make the scalar sector quasi-free at high energies—an explicit field-induced asymptotic freedom.
  • The current-current interaction generated by the averaging is a new, renormalizable, scale-dependent coupling that must be included in any effective description of charged scalars in fluctuating backgrounds.
  • The spectral density prediction (Lorentzian broadening with width ∝ Δ~) gives a concrete handle: quasi-particle lifetimes in the noisy background are controlled by the noise autocorrelation strength.
  • The noiseless limit recovers the standard λφ⁴ results (e.g. λ(t) = λ₀/(1 - 3λ₀t/16π²)), so the new effects are entirely attributable to the stochastic background.

Reading between the lines

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

  • Colored-noise check: Because the derivation leans on the δ-function autocorrelation, a noise with finite correlation time (as in real heavy-ion fields, ~1 fm/c) would produce a non-local current-current interaction; computing the same beta functions with, say, an exponential or Gaussian correlator would test whether the qualitative flow (complex couplings, pole-free strong-field regime) survives.
  • Mixing-term caveat: The paper explicitly drops the mixed λ-Δ~ vertex correction ('we shall ignore the mixing term in our present analysis', Sec. VII); including it would typically generate new UV structures, and one might expect the β-functions to acquire off-diagonal mixing that could alter the fixed-point structure.
  • Phenomenological translation: In heavy-ion collisions the in-plane electric field lasts ~1 fm/c; if the noise-induced width ∝ Δ~ is sizable, pion spectral functions in the fireball could show broadening beyond thermal effects, a measurable signature in dilepton or pion spectra.
  • The complex couplings imply the effective Lagrangian is not Hermitian after noise averaging; whether unitarity is restored in the n→0 replica limit or the theory is genuinely non-unitary (dissipative) is left implicit in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript studies a complex lambda-phi^4 scalar field coupled to a stochastic electric background. After replica averaging over Gaussian white-noise fluctuations of the gauge potential, Eq. (3), an effective current-current interaction (17) with dimensionless strength Delta_tilde = q^2 m^2 Delta is obtained. The authors compute one-loop self-energy and vertex corrections in weak- and strong-field limits, define counterterms through conditions (51)-(52), and derive beta functions for lambda and Delta_tilde, Eqs. (78)-(79) and (85)-(86). They claim that in the weak-field regime the noise acts as a damping factor and in the strong-field regime both couplings flow to zero, implying asymptotic freedom. The paper also presents spectral densities showing broadened quasi-particle states.

Significance. If the renormalization-group results were correct, the paper would establish a concrete mechanism by which classical electric-field fluctuations alter the running of couplings in a scalar effective theory relevant to pion-like fields in heavy-ion collisions. The replica-trick averaging is explicit, the weak-field propagator expansion and vertex integrals are carried out in appendices, and the paper provides analytic running couplings in both regimes. These are strengths. However, because the central counterterm computation is internally inconsistent, the quantitative RG flows are not established; the significance is therefore conditional on a successful rederivation.

major comments (4)
  1. [Sec. V A, Eqs. (51)-(55) and Eq. (62)] The counterterms do not enforce the stated renormalization conditions. From Eq. (53), partial Sigma/partial p^2 at p^2=m^2 equals +2i Delta_tilde/(4pi)^2 [2/epsilon - gamma + ln(4pi mu^2/m^2) + 1], so Eq. (52) requires delta Z of that sign, the opposite of Eq. (54). Moreover, substituting Eqs. (54)-(55) into Eq. (51) does not give zero; the finite (qE)^2 term is missing from delta m^2. Indeed Eq. (62) evaluated at p^2=m^2 leaves lambda (qE)^2/[12(4pi)^2 m^2] != 0. Because Eq. (77) is linear in delta Z, the sign error propagates into the beta functions (78)-(79), so the central RG result is not derived as stated.
  2. [Sec. V B, Eqs. (56)-(58)] The strong-field counterterms suffer the same sign inconsistency. The coefficient of (p^2-m^2) in Eq. (56) is +4i Delta_tilde (qE)/[(4pi)^2 m^2] L, so condition (52) requires delta Z of that sign, not the negative sign shown in Eq. (57). The delta m^2 in Eq. (58) likewise does not match the mass condition. Consequently the statement in Sec. VI B that the renormalized self-energy vanishes, and the beta functions (85)-(86) obtained from Eq. (57), are not justified.
  3. [Sec. V A, Eq. (53) vs. Appendix B] The factorized expression for the self-energy does not reproduce the explicit integrals (B9) and (B14): the lambda finite electric-field term has the opposite sign, and the Delta_tilde finite term proportional to (qE)^2 is absent from Eq. (53). Since Eqs. (54)-(55) are derived from Eq. (53), the counterterms are not based on the computed self-energy. This needs to be reconciled before any RG flow can be trusted.
  4. [Sec. VIII B, Eqs. (87)-(88) and Appendix E2] Solving Eq. (86) with Eq. (87) gives lambda(t) = lambda_0 Delta_tilde(t)/Delta_tilde_0 = lambda_0/(1 + i c Delta_tilde_0 t), not lambda_0 Delta_tilde_0/Delta_tilde(t) as stated in Eq. (88). As written, Eq. (88) implies |lambda(t)| grows linearly with t, contradicting the text that both couplings run as ~1/t and the conclusion of asymptotic freedom. The strong-field running must be re-solved after the counterterm sign is corrected.
minor comments (4)
  1. [Sec. VIII A, Eq. (84)] The lambda Landau-pole condition should be t < 16 pi^2/(3 lambda_0), not 3 lambda_0/(16 pi^2); as written the second entry in the min is dimensionally inconsistent.
  2. [Sec. I and Eq. (80)] The definition of the logarithmic scale t differs: the Introduction says t = ln(p^2/m^2), while Eq. (80) defines t = (1/2) ln(p^2/mu^2). This should be unified since the beta functions and Landau-pole locations depend on the convention.
  3. [Sec. VI A, Fig. 3] The numerical value 'lambda = 2,06' should use a decimal point, and the horizontal axis label p0 [MeV] appears inconsistent with m_pi = 140 MeV if p0 is intended to be on-shell energy; please clarify the kinematic variable.
  4. [General] The white-noise model of Eq. (3) is a strong physical idealization; a colored-noise background with finite correlation time would produce a non-local effective interaction and different beta functions. This should be stated as a limitation rather than presented as the generic outcome for heavy-ion fields.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and Δ is an input, not a fitted output.

full rationale

The paper's chain is: (i) specify a Gaussian white-noise model for δA (Eqs. (2)–(4)); (ii) average the generating functional over this noise using the replica trick (Eqs. (12)–(14)), which produces the current–current term (Eq. (17)); (iii) use standard Schwinger propagators in weak/strong E regimes (Eqs. (26), (30), (31)–(32)); (iv) compute one-loop self-energy (Eqs. (53), (56)) and vertex functions (Eqs. (69), (70)); (v) define counterterms through the renormalization conditions (51)–(52) and vertex conditions (72), then obtain beta functions (78)–(79), (85)–(86) by Eq. (77). At no point is a target result inserted as an input. Δ is a free parameter characterizing the noise correlator, not fitted to the quantities later 'predicted'; the beta functions are computed, not assumed. Citations [13–16] supply the averaging technique (replica trick plus Gaussian integration), but the replica trick itself is cited to standard work [18], and the target quantities (current–current vertex, self-energy, vertex corrections, RG flows) are derived explicitly in this paper. The noiseless limit correctly reduces to the standard λφ^4 one-loop beta function (Eq. (83)), an independent consistency check. A possible algebraic error in the counterterms (sign/magnitude of δZ, δm² in Eqs. (54)–(55) vs. conditions (51)–(52)) would be a correctness defect, not a circularity: an inconsistent calculation is not one that assumes its conclusion. Therefore no circular step is established.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central calculation has two genuine inputs: Δ, the noise correlation, and the Gaussian white-noise distribution. Everything else (Schwinger propagator, dimensional regularization, one-loop perturbation theory) is standard. There are no fitted parameters and no new particles or forces.

free parameters (1)
  • Δ (noise auto-correlation strength)
    Introduced in Eq. (3) as the variance of Gaussian white noise on the background gauge field; after replica averaging it becomes the effective current-current coupling Δ~=q²m²Δ. Its value is not derived or fitted; numerical choices appear only in Fig. 3.
assumptions (4)
  • domain assumption The stochastic fluctuations are Gaussian white noise with ⟨δAμ(x)δAν(y)⟩=Δ δμν δ⁴(x-y).
    This is the input noise model; the replica average and the resulting local current-current interaction depend directly on it (Eq. 3).
  • domain assumption The replica-trick identity ln Z = lim_{n→0}(Z^n-1)/n applies and can be combined with standard perturbative renormalization.
    Used in Eq. (13); the n→0 limit is not explicitly implemented in the Feynman rules, and the paper assumes the quenched average commutes with renormalization.
  • domain assumption One-loop perturbation theory in λ and Δ~ is valid in both weak and ultra-strong electric field regimes.
    The paper computes only one-loop self-energies and vertex corrections and does not discuss convergence, especially for |qE|/m²≫1.
  • domain assumption In the ultra-strong field regime the propagator is approximated by the Landau-level l=0 mode of Eq. (32).
    The strong-field calculations in Appendices B and D retain only the lowest Landau level.

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Pith. "Pith review of Electric field fluctuations and renormalization group flows in a self-interacting scalar field theory." pith.science (2026). https://pith.science/paper/ZM6UIXTJ

@misc{pith2026260716932,
  author       = {Pith},
  title        = {Pith review of: Electric field fluctuations and renormalization group flows in a self-interacting scalar field theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZM6UIXTJ}},
  note         = {Machine review of arXiv:2607.16932}
}
read the original abstract

We consider a self interacting charged scalar field represented by the complex {\lambda}{\phi}4 model, embedded in a background electric field exhibiting classical stochastic fluctuations. We studied the effects of the classical stochastic noise on the physical parameters of the scalar field theory, for both weak and ultra strong electric field regimes. The stochastic background electric field is included in the Schwinger propagator through the covariant derivative, and the generating functional of the theory is found by means of the replica trick in order to compute the statistical average over electric fluctuations. As a result of the averaging process, an effective interaction between charged currents emerges, with a coupling constant proportional to the magnitude of the auto-correlation function of the electric field fluctuations. We obtained the dressed propagators, the interaction vertices, and the renormalization group equations of the theory, along with the corresponding streamplots in the manifold of interaction couplings.

Figures

Figures reproduced from arXiv: 2607.16932 by the authors.

Figure 2
Figure 2. FIG. 2: Self energy diagram at first order in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 1
Figure 1. FIG. 1: Feynman diagram for the first order in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Spectral density for the bare and dressed [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Diagrams contributing to the 4-point vertex am [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Feynman rule for the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: s-channel diagram for the [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Stream plot of the beta functions in the very weak electric field regime [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

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Reference graph

Works this paper leans on

26 extracted references · 1 linked inside Pith

  1. [1]

    Baker and F

    M. Baker and F. Zachariasen, Pion-pion scattering in the ϕ4 theory, Phys. Rev.118, 1659 (1960)

    1960

  2. [2]

    Busza, K

    W. Busza, K. Rajagopal, and W. van der Schee, Heavy ion collisions: The big picture and the big questions, Annual Review of Nuclear and Particle Science68, 339 (2018)

    2018

  3. [3]

    Iancu, Qcd in heavy ion collisions (2012), arXiv:1205.0579 [hep-ph]

    E. Iancu, Qcd in heavy ion collisions (2012), arXiv:1205.0579 [hep-ph]

    arXiv 2012

  4. [4]

    Deng and X.-G

    W.-T. Deng and X.-G. Huang, Electric fields and chiral magnetic effect in cu+au collisions, Physics Letters B 742, 296 (2015)

    2015

  5. [5]

    Cheng, S

    Y.-L. Cheng, S. Zhang, Y.-G. Ma, J.-H. Chen, and C. Zhong, Electromagnetic field from asymmetric to sym- metric heavy-ion collisions at 200 GeV/c, Phys. Rev. C 99, 054906 (2019)

    2019

  6. [6]

    Voronyuk, V

    V. Voronyuk, V. D. Toneev, S. A. Voloshin, and W. Cass- ing, Charge-dependent directed flow in asymmetric nu- clear collisions, Phys. Rev. C90, 064903 (2014)

    2014

  7. [7]

    Hirono, M

    Y. Hirono, M. Hongo, and T. Hirano, Estimation of the electric conductivity of the quark gluon plasma via asym- metric heavy-ion collisions, Phys. Rev. C90, 021903(R) (2014)

    2014

  8. [8]

    S. N. Alam, V. Roy, S. Ahmad, and S. Chattopadhyay, Electromagnetic field fluctuation and its correlation with the participant plane in Au + Au and isobaric collisions at √sN N= 200 GeV, Phys. Rev. D104, 114031 (2021)

    2021

Show all 26 references
  1. [9]

    Gao, Y.-L

    Y. Gao, Y.-L. Guo, L. Zhang, G.-C. Yong, Z.-Y. Liu, and W. Zuo, Pion production and absorption in heavy- ion collisions, Phys. Rev. C104, 044607 (2021)

    2021

  2. [10]

    Adamczewski-Musch, O

    J. Adamczewski-Musch, O. Arnold, C. Behnke, A. Be- lounnas, A. Belyaev, J. C. Berger-Chen, A. Blanco, C. Blume, M. B¨ ohmer, P. Bordalo, S. Chernenko, L. Chlad, I. Ciepa l, C. Deveaux, J. Dreyer, E. Epple, L. Fabbietti, O. Fateev, P. Filip, P. Fonte, C. Franco, J. Friese, I. F...

    2020

  3. [11]

    Kumar, K

    A. Kumar, K. Gowthama, V. Chandra, and S. Dash, Electric, thermal, and thermoelectric response of a hot pion gas in a time dependent background magnetic field, Phys. Rev. D112, 096026 (2025)

    2025

  4. [12]

    Schwinger, On gauge invariance and vacuum polariza- tion, Phys

    J. Schwinger, On gauge invariance and vacuum polariza- tion, Phys. Rev.82, 664 (1951)

    1951

  5. [13]

    J. D. Casta˜ no Yepes, M. Loewe, E. Mu˜ noz, J. C. Rojas, and R. Zamora, Qed fermions in a noisy magnetic field background, Phys. Rev. D107, 096014 (2023)

    2023

  6. [14]

    J. D. Casta˜ no Yepes and E. Mu˜ noz, Fermion self-energy and effective mass in a noisy magnetic background, Phys. Rev. D110, 056003 (2024)

    2024

  7. [15]

    J. D. Casta˜ no Yepes and E. Mu˜ noz, Exploring magnetic fluctuation effects in qed gauge fields: Implications for mass generation, Phys. Rev. D109, 056007 (2024)

    2024

  8. [16]

    J. D. Casta˜ no Yepes and E. Mu˜ noz, Anisotropic photon and dilepton yield in a thermalized quark-gluon plasma under spatial magnetic fluctuations, Phys. Rev. D111, 076028 (2025)

    2025

  9. [17]

    Loewe, D

    M. Loewe, D. Valenzuela, and R. Zamora, Catalysis and inverse electric catalysis in a scalar theory, Phys. Rev. D 105, 036017 (2022)

    2022

  10. [18]

    M´ ezard, G

    M. M´ ezard, G. Parisi, M. A. Virasoro, and D. J. Thouless, Spin Glass Theory and Beyond(1987)

    1987

  11. [19]

    Dittrich and M

    W. Dittrich and M. Reuter,Effective Lagrangians in quantum electrodynamics(Springer, 1985)

    1985

  12. [20]

    Loewe, D

    M. Loewe, D. Valenzuela, and R. Zamora, Effective po- tential and mass behavior of a self-interacting scalar field theory due to thermal and external electric and magnetic fields effects, The European Physical Journal A59, 184 (2023). Appendix A: The propagator for a very weak...

    2023

  13. [21]

    Case (A): V ery weak field limit|qE|/m 2 ≪1 The self energy contribution arising from theλϕ 4 interaction is given by Σλ(E) =− λ 2 Z d4k (2π)4 ( 1 k2 +m 2 + (qE)2 " −1 (k2 +m 2)3 + 2k2 ∥ (k2 +m 2)4 #) .(B3) The first integral is logarithmically divergent, and can be evaluated ...

  14. [22]

    Case (B): V ery strong field limit|qE|/m 2 ≫1 In this case, we compute the self-energy contributions by using the very strong limit form of the propagator, Eq. (32). Σλ(p, E) =− λ 2 Z d4k (2π)4 DE(k) =− λ 2 2 Z d2k∥ (2π)2 e− k2 ∥ qE Z d2k⊥ (2π)2 1 k2 ⊥ +m 2 E =−λ πqE (2π)2 µϵ ...

  15. [23]

    −1 (k2 +m 2)3 + 2k2 ∥ (k2 +m 2)4 #) × ( 1 (p+k) 2 +m 2 + (qE)2

    Case (A): V ery weak field limit|qE|/m 2 ≪1 As discussed in the main text, thes−channel contribution to the vertex function is given by the integral iV(p, E) = Z d4k (2π)4 iDE(k)iD E(p+k).(D1) 17 Using the propagator for the weak field limit, given by Eq. (30), the vertex func...

  16. [24]

    Case (B): V ery strong field limit|qE|/m 2 ≫1 In the strong field case we use the propagator from Eq. (32). Then the vertex function results iV(p, E) = Z d4k (2π)4 2e− k2 ∥ qE k2 ⊥ +qE+m 2 2e− (p∥ +k∥) 2 qE (p⊥ +k ⊥)2 +qE+m 2 .(D14) 18 Separating the integral in the perpendicu...

  17. [25]

    (E1) and Eq

    W eak electric field regime In this section, we consider the analytical solution to the system of differential equations for the couplingsg= (λ, ˜∆), β ˜∆ = 2 π2 ˜∆2 1− i 4 ,(E1) βλ = 3 16π2 λ2 − i 2π2 λ ˜∆.(E2) We first notice that the first two equations, Eq. (E1) and Eq. (E...

  18. [26]

    V ery strong electric field regime The beta functions for the couplings and the physical mass parameters are given by β ˜∆ =−i qE π2m2 ˜∆2,(E17) βλ =−i qE π2m2 λ ˜∆ (E18) The first two equations can be solved by direct integration. The solution to the ˜∆ equation can be obtain...

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