Pith. sign in

REVIEW 3 major objections 5 minor 54 references

In Lorentz- and CPT-violating QED, multiplying the fermion kinetic term by the entire function e^{-/D/Λ} makes the one-loop Carroll-Field-Jackiw coefficient finite and unambiguous: C = (3e²/16π²) f(m/Λ), with no external UV regulator.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 03:29 UTC pith:AGSZD7LT

load-bearing objection A real one-loop CFJ computation with a plausible finite deformation, but the Wick rotation that guarantees finiteness is asserted, not proved. the 3 major comments →

arxiv 2607.13858 v1 pith:AGSZD7LT submitted 2026-07-15 hep-th

Induced CFJ Term in nonlocal Lorentz-Violating QED: Natural Regularization from Entire Dirac Operators

J. R. Nascimento , A. Yu. Petrov , P. J. Porfirio , Ramires N. da Silva This is my paper
classification hep-th PACS 11.30.Cp11.10.Gh
keywords nonlocal QEDLorentz violationCPT violationCarroll-Field-Jackiw termentire functionDirac operatorradiative correctionsUV regularization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper studies a nonlocal version of Lorentz- and CPT-violating QED in which the fermion kinetic term is multiplied by an entire function of the Dirac operator. It claims that the radiatively induced Carroll-Field-Jackiw (CFJ) term keeps its usual tensor structure but acquires a finite coefficient without any external UV regulator: for f(/D)=e^{-/D/Λ}, after Wick rotation the loop integral is exponentially suppressed and gives C = (3e²/16π²) f(μ), where μ=m/Λ and f is a real, monotonically decreasing deformation function. For the oscillatory form factor e^{-i/D/Λ}, the coefficient is also finite but only in the sense of Abel regularization, and it can change sign as μ grows. The reason this matters is that the long-standing regularization ambiguity of the local CFJ coefficient is converted into a controlled, scale-ratio-dependent prediction.

Core claim

The paper's central claim is that nonlocality built from entire functions of the Dirac operator acts as an intrinsic regulator for the one-loop CFJ term in Lorentz/CPT-violating QED. Specifically, for f1(/D)=e^{-/D/Λ}, the Wick-rotated one-loop integrand is absolutely convergent, and the induced coefficient is C = C_loc f(μ), with C_loc = 3e²/16π² and f(μ) a real dimensionless deformation function that decreases monotonically and equals 1 at μ ≃ 0.127. For f2(/D)=e^{-i/D/Λ}, the Wick-rotated integral is oscillatory; with an Abel-damping prescription it is conditionally convergent, yields a finite f(μ) that can become negative for larger μ, and reduces to the same local coefficient if the non

What carries the argument

The central object is the entire nonlocal form factor f(/D) multiplying the Dirac operator in the fermion Lagrangian. Choosing f(/D)=e^{-/D/Λ} gives momentum-space factors that make the one-loop denominator positive definite and the integrand exponentially damped. The calculational machinery is the Fréchet (Duhamel) expansion of the operator exponential in powers of the gauge field, which produces ordered linear and quadratic vertices; a derivative expansion isolates the parity-odd trace; and a Wick rotation performed at the level of the Dirac operator converts the Minkowski integral into the Euclidean master formula C = C_loc f(μ). The same machinery, with an Abel damping factor e^{-ηρ}, ha

Load-bearing premise

The load-bearing assumption is that the Wick rotation from Minkowski to Euclidean momenta can be performed on the nonlocal one-loop integral, at the level of the Dirac operator, without picking up extra contributions from complex-plane singularities; the paper justifies this only a posteriori from exponential damping, and if the contour rotation receives pole contributions the claimed finite and unambiguous coefficient changes.

What would settle it

A direct numerical evaluation of the original Minkowski-space loop integral, along a contour with a small energy prescription and without invoking the Euclidean rotation, would settle the issue: if the result differs from C = C_loc f(μ), the claimed finite value is an artifact of the rotation. Equivalently, checking whether the kernel has complex-plane poles in the quadrants crossed by the clockwise rotation would reveal any missing contribution.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For f(/D)=e^{-/D/Λ}, the CFJ coefficient is finite and real for all m/Λ > 0, with no external UV cutoff, dimensional regularization, or Pauli-Villars regulator needed.
  • The CFJ tensor structure is preserved; nonlocality only rescales the coefficient by f(μ), so observable signatures of the CFJ term persist but with a modified strength.
  • Large fermion mass relative to the nonlocality scale suppresses the induced coefficient, and for the oscillatory form factor the coefficient can flip sign.
  • The formal order of taking the local limit and performing the loop integral matters: keeping the nonlocal regulator and integrating first selects a definite finite value, whereas sending Λ→∞ first reproduces the regularization-dependent local result.
  • Contact (two-photon) vertices in these nonlocal theories do not contribute to the parity-odd CFJ structure; only the diagram with two linear vertices does.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the calculation holds, the ratio m/Λ acts as a physical dial on the CFJ coefficient, so precision measurements of vacuum birefringence or CMB polarization rotation could in principle constrain Λ for a given fermion mass, since the induced effect is suppressed when m ≫ Λ.
  • The same entire-function regulator strategy could be applied to other radiatively induced Lorentz-violating or topological terms, suggesting a general pattern: entire functions of the kinetic operator define finite, scale-dependent coefficients without introducing new poles.
  • A testable cross-check would be to derive the same CFJ coefficient using an independent regularization on the nonlocal theory and compare the μ-dependence of f(μ); universality across regulators would strengthen the claim that the ambiguity is genuinely resolved.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the one-loop radiative generation of the Carroll-Field-Jackiw (CFJ) term in a nonlocal Lorentz- and CPT-violating extension of QED, where the fermion kinetic term is dressed by an entire function of the Dirac operator. After deriving the one-loop effective action using a Fréchet expansion of the nonlocal vertices, the authors compute the parity-odd contribution for two representative form factors: f(/D)=exp(-/D/Λ) and f(/D)=exp(-i/D/Λ). For the first case, they claim that after Wick rotation the loop integral is exponentially convergent and yields a finite CFJ coefficient C=C_loc f(m/Λ), with C_loc=3e^2/16π^2 and f(μ) defined in Eq. (37). For the second case, the integral is oscillatory and is rendered finite by an Abel regularization. The central assertion is that entire-function nonlocality acts as an intrinsic UV regulator, preserving the CFJ tensor structure while continuously deforming its coefficient by the mass-to-nonlocality ratio.

Significance. If the derivation is correct, the paper makes a useful contribution: it provides a concrete class of nonlocal LV QED models in which the radiatively induced CFJ coefficient is finite without an external UV regulator, and it exhibits a smooth, calculable dependence on m/Λ. The paper's strengths include a clear presentation of the nonlocal vertex structure via the Fréchet expansion, explicit recognition of the noncommutativity of the local limit and loop integration, and numerical plots of the resulting deformation function. However, the central result rests on a Wick-rotation step that is asserted rather than proved, and the reduction from the Dirac trace to the one-dimensional integral is not shown. These are load-bearing gaps, so the significance is currently conditional on closing them.

major comments (3)
  1. [§III.A, Eqs. (33)–(37)] The Wick rotation from Eq. (33) to Eq. (37) is the central technical step and is not justified. The denominator in Eq. (33), Δ(ρ)=ρ²−μ²+2iμρ sinρ, is an entire function whose zeros are never analyzed. For μ=nπ there are actual poles on the real integration contour at ρ=nπ; for generic μ, complex zeros could be crossed when the integration path is rotated. The text says the rotation is performed 'at the level of the Dirac operator, prior to rationalizing the propagator', but Eq. (33) is already a rationalized, one-dimensional integral. The a posteriori damping argument after Eq. (38) concerns the Euclidean integrand and does not prove that the original contour deformation is free of singularities. Moreover, the Minkowski integrand decays only as O(1/ρ), so the usual 'arc at infinity gives zero' argument is not available by absolute convergence. Unless a detailed contour analysis is suppli
  2. [§III.A, Eqs. (30)–(33)] The reduction of the trace expression in Eq. (29) to the compact radial integral in Eq. (33) is a black box. The text says 'applying the symmetric tensor reduction ... and isolating the coefficient', but the angular integrations and the relative coefficients of the five tensor structures in Eq. (31) are not shown. Since the final coefficient C_loc=3e²/16π², the sign of the CFJ term, and the function f(μ) all depend on these factors, this is not a cosmetic omission. The authors should include an appendix (or at least an intermediate expression after angular integration) so that the result can be independently checked.
  3. [§III.B, Eqs. (44)–(48)] For the oscillatory form factor, the Abel regularization is introduced but its uniqueness is not established. The kernel in Eq. (44) is only conditionally convergent; Abel summation is one of several summation methods that can assign different values to such integrals. The paper claims a 'well-defined finite CFJ coefficient', but Fig. 4 is plotted for η=0.03 rather than the η→0⁺ limit of Eq. (48). To support the uniqueness claim, the authors should show that the Abel limit exists and is independent of the regularization path, or clearly state that the result is prescription-dependent. This does not affect Case 1, but it is load-bearing for the Case-2 claim.
minor comments (5)
  1. [§III.A, after Eq. (14)] The phrase 'the contact term involving.' is incomplete; it should presumably read 'the contact term involving V2'.
  2. [After Eq. (37)] Typo: 'deformation fuction' should be 'deformation function'.
  3. [Fig. 4 caption] The caption states η=0.03, but Eq. (48) defines f(μ) as the η→0⁺ limit. Please clarify whether the figure shows the limiting function or a finite-η approximation.
  4. [Reference [27]] The author name appears corrupted as 'Rachwa/suppress l'. Please correct.
  5. [General] The expansion in Eq. (25) uses a Clifford-algebra basis without defining the normalization of the higher-rank components; a brief definition would improve readability.

Circularity Check

0 steps flagged

No significant circularity: f(μ) is an explicit integrated deformation, not a fit; self-citations are confined to model setup.

full rationale

The central equation C=C_loc f(μ) is not circular: Eq. (33) is an explicit one-loop integral, Eq. (37) is a specific Euclidean integral defining f(μ), and C_loc=3e²/16π² is quoted from the independent reference [17]. No parameter is fitted to the target coefficient, so the relation is a rescaling of a computed integral, not a definitional identity. The form factors f1=e^{-/D/Λ} and f2=e^{-i/D/Λ} are adopted from the authors' own Ref. [30] ('we analyze two representative exponential choices introduced in Ref. [30]'), but they serve as the model ansatz whose consequences are computed in this paper; the CFJ result is not imported from that citation. The one load-bearing caveat is the Wick rotation from Eq. (33) to Eqs. (35)/(37). The paper itself concedes that 'analytic continuation to Euclidean space can be subtle' and supports the rotation only by the large-momentum decay of the Euclidean integrand (Eq. (38)), which shows the contour receives no contribution from infinity but does not exclude finite complex-plane poles or conditional-convergence obstructions. This is an omitted proof / correctness risk, not a reduction of the result to its inputs. Hence no circular step is established.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

No new particles, forces, or conserved quantities are introduced. The nonlocal form factor f(/D) and the scale Λ are new structures but are inherited from ref. [30] and do not have independent falsifiable handles of their own beyond the model's predictions.

free parameters (2)
  • nonlocality scale Λ (or ratio μ=m/Λ) = unfixed
    New scale introduced by the model; not determined by any data or principle in the paper. The central result C = C_loc f(μ) is parameterized by μ.
  • Abel regularization parameter η = 0.03 in Figures 4 and 5; formal limit η→0+ in Eq. (48)
    Chosen by hand for the shown numerical evaluation; convergence and uniqueness of the η→0 limit is not demonstrated.
axioms (5)
  • domain assumption The derivative expansion (Eq. 28), truncated at first order, captures the full CFJ contribution
    Used to go from Eq. (29) to Eq. (31); assumes the external field varies slowly relative to loop momentum.
  • domain assumption Symmetric tensor reduction p_μ p_ν → p²/4 η_μν is valid in the nonlocal integrals
    Applied to isolate the CFJ coefficient in Eq. (33); standard only when a Lorentz-preserving regulator is implicit, which is not specified here.
  • ad hoc to paper Wick rotation at the level of the Dirac operator, prior to rationalizing the propagator, gives the correct Euclidean effective action
    Nonstandard prescription; the paper notes analytic-continuation subtleties for nonlocal theories (refs [47-49]) and justifies it only a posteriori.
  • ad hoc to paper Abel regularization assigns the physical value to the oscillatory Case-2 integral
    Eq. (45); the choice of Abel summability over other summability methods is not physically motivated.
  • domain assumption The V2 contact term does not project onto the local CFJ operator
    Argued from Bose symmetry of Eq. (17), but the full calculation showing the vanishing is not exhibited.

pith-pipeline@v1.3.0-alltime-deepseek · 11932 in / 13278 out tokens · 125260 ms · 2026-08-02T03:29:11.391458+00:00 · methodology

0 comments
read the original abstract

We investigate the radiative generation of the Carroll--Field--Jackiw (CFJ) term in nonlocal Lorentz- and CPT-violating extensions of quantum electrodynamics. Nonlocality is introduced through entire nonlocal form factors being functions of the Dirac operator in the fermionic sector. Using the derivative expansion of the one-loop fermionic determinant, we derive the induced CFJ contribution and analyze its ultraviolet (UV) behavior after Wick rotation. For one class of nonlocal form factors, loop integrals become naturally convergent, yielding a finite CFJ coefficient without the need for an external UV regulator. For a second class, the coefficient is rendered finite through an Abel regularization prescription. In both cases, the CFJ structure is preserved while its coefficient is continuously deformed by the ratio between the fermion mass and the nonlocality scale. Our results demonstrate that nonlocality provides a natural framework for controlling UV contributions to radiatively induced topological terms.

Figures

Figures reproduced from arXiv: 2607.13858 by A. Yu. Petrov, J. R. Nascimento, P. J. Porfirio, Ramires N. da Silva.

Figure 1
Figure 1. Figure 1: One-loop contribution to the induced CFJ term in th [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Dimensionless nonlocal deformation function [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Four plots of Euclidean CFJ coefficient C ′ = C/e 2 as a function of the dimensionless momentum variable ρ, for different µ values: µ = 2 (red line), µ = 1 (green line), µ = 0.5 (yellow line) and µ = 0.25 (blue line). 12 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Dimensionless nonlocal deformation function [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Abel-regularized (for η = 0.03) Euclidean CFJ coefficient C ′ = C/e 2 as a function of the dimensionless variable ρ, for different µ values: µ = 1 (green line), µ = 0.5 (yellow line) and µ = 0.25 (blue line). IV. CONCLUSIONS We studied a nonlocal Lorentz- and CPT-violating extension of QED in which the fermionic kinetic operator is dressed by an entire function of the gauge-covariant Dirac operator. Our fo… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

54 extracted references · 25 linked inside Pith

  1. [1]

    M. E. Peskin and D. V. Schroeder, An Introduction to Quant um Field Theory, (Addison- Wesley, 1995)

    1995

  2. [2]

    Srednicki, Quantum Field Theory, (Cambridge Univers ity Press, 2007)

    M. Srednicki, Quantum Field Theory, (Cambridge Univers ity Press, 2007)

    2007

  3. [3]

    Hanneke, S

    D. Hanneke, S. Fogwell, G. Gabrielse, Phys. Rev. Lett. 10 0, 120801 (2008)

    2008

  4. [4]

    M. B. Green and J. H. Schwarz and E. Witten, Superstring Th eory, (Cambridge University Press, (1987)

    1987

  5. [5]

    Polchinski, String Theory, (Cambridge University Pr ess, 1998)

    J. Polchinski, String Theory, (Cambridge University Pr ess, 1998)

    1998

  6. [6]

    H. Gies, J. Jaeckel, Phys. Rev. Lett. 93, 110405 (2004). a rXiv:hep-ph/0405183 [hep-ph]

    Pith/arXiv arXiv 2004

  7. [7]

    Amelino-Camelia, Introduction to Quantum-Gravity P henomenology, 669, 59-100, (Lect

    G. Amelino-Camelia, Introduction to Quantum-Gravity P henomenology, 669, 59-100, (Lect. Notes Phys. 2005)

    2005

  8. [8]

    V. A. Kosteleck´ y, Phys. Rev. D, 69 (2004) 105009 [arXiv:hep-th/0312310 [hep-th]]

    Pith/arXiv arXiv 2004

  9. [9]

    Mattingly, Modern Tests of Lorentz Invariance, Livin g Rev

    D. Mattingly, Modern Tests of Lorentz Invariance, Livin g Rev. Relativity, 8, 5, (2005) [arXiv:gr-qc/0502097 [gr-qc]]

    Pith/arXiv arXiv 2005

  10. [10]

    V. A. Kostelecky, C. D. Lane and A. G. M. Pickering, Phys. Rev. D 65 (2002), 056006 [arXiv:hep-th/0111123 [hep-th]]

    Pith/arXiv arXiv 2002

  11. [11]

    J. M. Chung, Phys. Rev. D 60 (1999), 127901 [arXiv:hep-th/9904037 [hep-th]]

    Pith/arXiv arXiv 1999

  12. [12]

    A. A. Andrianov, P. Giacconi and R. Soldati, JHEP 02 (2002), 030 [arXiv:hep-th/0110279 [hep-th]]. 17

    Pith/arXiv arXiv 2002

  13. [13]

    Perez-Victoria, JHEP 04 (2001), 032 [arXiv:hep-th/0102021 [hep-th]]

    M. Perez-Victoria, JHEP 04 (2001), 032 [arXiv:hep-th/0102021 [hep-th]]

    Pith/arXiv arXiv 2001

  14. [14]

    Altschul, Phys

    B. Altschul, Phys. Rev. D 70 (2004), 101701 [arXiv:hep-th/0407172 [hep-th]]

    Pith/arXiv arXiv 2004

  15. [15]

    A. F. Ferrari, J. R. Nascimento and A. Y. Petrov, Eur. Phy s. J. C 80 (2020) no.5, 459 [arXiv:1812.01702 [hep-th]]

    Pith/arXiv arXiv 2020

  16. [16]

    S. M. Carroll, G. B. Field and R. Jackiw, Phys. Rev. D 41 (1990), 1231

    1990

  17. [17]

    Jackiw, Int

    R. Jackiw, Int. J. Mod. Phys. B 14 (2000), 2011-2022 [arXiv:hep-th/9903044 [hep-th]]

    Pith/arXiv arXiv 2000

  18. [18]

    O. A. Battistel and G. Dallabona, Nucl. Phys. B 610, 316 (2001) [arXiv:hep-th/0012181 [hep-th]]

    Pith/arXiv arXiv 2001

  19. [19]

    Bonneau, Nucl

    G. Bonneau, Nucl. Phys. B 593, 398 (2001) [arXiv:hep-th/0008210 [hep-th]]

    Pith/arXiv arXiv 2001

  20. [20]

    Altschul, Phys

    B. Altschul, Phys. Rev. D 99 (2019) no.12, 125009 [arXiv:1903.10100 [hep-th]]

    Pith/arXiv arXiv 2019

  21. [21]

    formulated key ideas of this concept. Further, nonlocal operat ors defined through en- tire functions have been investigated in various contexts, including modified gauge theories, quantum gravity, higher-derivative models and superfield theories (see, e.g., [ 22–28] and references therein). Entire functions can suppress UV contribu tions without introducing...

  22. [22]

    Biswas, A

    T. Biswas, A. Mazumdar and W. Siegel, JCAP 03 (2006), 009 [arXiv:hep-th/0508194 [hep- th]]

    Pith/arXiv arXiv 2006

  23. [23]

    G. V. Efimov, Commun. Math. Phys. 5 (1967) no.1, 42-56

    1967

  24. [24]

    Superrenormalizable gauge and gravi tational theories,

    E. T. Tomboulis, “Superrenormalizable gauge and gravi tational theories,” [arXiv:hep- th/9702146 [hep-th]]

    arXiv

  25. [25]

    Biswas, T

    T. Biswas, T. Koivisto and A. Mazumdar, JCAP 11 (2010), 008 [arXiv:1005.0590 [hep-th]]

    Pith/arXiv arXiv 2010

  26. [26]

    Modesto and L

    L. Modesto and L. Rachwal, Nucl. Phys. B 889 (2014), 228-248 [arXiv:1407.8036 [hep-th]]

    Pith/arXiv arXiv 2014

  27. [27]

    Modesto, Phys

    L. Modesto, Phys. Rev. D 86 (2012), 044005 [arXiv:1107.2403 [hep-th]]

    Pith/arXiv arXiv 2012

  28. [28]

    F. S. Gama, J. R. Nascimento, G. Olmo, A. Y. Petrov and P. P orf´ ırio, Phys. Lett. B879 (2026), 140685 [arXiv:2602.18237 [hep-th]]

    Pith/arXiv arXiv 2026

  29. [29]

    Modesto and L

    L. Modesto and L. Rachwa/suppress l, Int. J. Mod. Phys. D26 (2017) no.11, 1730020

    2017

  30. [30]

    J. R. Nascimento, G. J. Olmo, A. Yu. Petrov, P. J. Porf´ ır io and R. N. Silva, Phys. Rev. D 113, 045015 (2026) [arXiv: 2505.07483 [hep-th]]

    arXiv 2026

  31. [31]

    J. W. Moffat, Phys. Rev. D 41 (1990), 1177-1184

    1990

  32. [32]

    C. D. Carone, Phys. Rev. D 102 (2020) no.9, 095006 [arXiv:2008.10525 [hep-ph]]

    Pith/arXiv arXiv 2020

  33. [33]

    Altschul, Phys

    B. Altschul, Phys. Rev. D 72 (2005), 025001 [arXiv:hep-th/0412113 [hep-th]]

    Pith/arXiv arXiv 2005

  34. [34]

    Colladay and V

    D. Colladay and V. A. Kostelecky, Phys. Rev. D 58 (1998), 116002 [arXiv:hep-ph/9809521 [hep-ph]]

    Pith/arXiv arXiv 1998

  35. [35]

    Colladay and V

    D. Colladay and V. A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997) [arXiv: hep-ph/9703464]

    Pith/arXiv arXiv 1997

  36. [36]

    Blanes, F

    S. Blanes, F. Casas, J. A. Oteo, and J. Ros, The Magnus exp ansion and some of its applica- tions, Phys. Rep. 470, 151 (2009)

    2009

  37. [37]

    B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics Vol. 222, Springer, New York ( 2003). 18

    2003

  38. [38]

    J. E. Campbell, Proceedings of the London Mathematical Society, 28, 381-390, (1897)

  39. [39]

    H. F. Baker, Proceedings of the London Mathematical Soc iety, 3, 24-47, (1905)

    1905

  40. [40]

    Cartan, Differential Calculus on Normed Spaces, (Inde pendent Publishing Platform, 2017)

    H. Cartan, Differential Calculus on Normed Spaces, (Inde pendent Publishing Platform, 2017)

    2017

  41. [41]

    Hausdorff, Berichte der Verhandlungen der K¨ oniglich S¨ achsischen Gesellschaft der Wis- senschaften zu Leipzig, 58, 19-48, (1906)

    F. Hausdorff, Berichte der Verhandlungen der K¨ oniglich S¨ achsischen Gesellschaft der Wis- senschaften zu Leipzig, 58, 19-48, (1906)

    1906

  42. [42]

    N. J. Higham, Functions of Matrices: Theory and Computation , SIAM, Philadelphia (2008)

    2008

  43. [43]

    Fr´ echet, Sur la notion de diff´ erentielle, (Nouvelles Annales de Math´ ematiques, 1911)

    M. Fr´ echet, Sur la notion de diff´ erentielle, (Nouvelles Annales de Math´ ematiques, 1911)

    1911

  44. [44]

    N. J. Higham and S. D. Relton, Higher order Fr´ echet derivatives of matrix functions and the level-2 condition number, SIAM J. Matrix Anal. Appl. 35, 1019 (2014)

    2014

  45. [45]

    A. H. Al-Mohy and N. J. Higham, Computing the Fr´ echet derivative of the matrix exponential, with an application to condition number estimation, SIAM J. Matrix Anal. Appl. 30, 1639 (2009)

    2009

  46. [46]

    ( 28) acts only on the momentum dependence of the free propagator

    for a comprehensive discussion of the method), based on the rela tion: /A(x) → /A ( x −i ∂ ∂p ) = /A(x) −i∂µ/A(x) ∂ ∂pµ + · · · (28) Because ∂µ(γ5/b) = 0, the derivative expansion in Eq. ( 28) acts only on the momentum dependence of the free propagator. Keeping terms up to first or der in derivatives and isolating the pieces relevant for the CFJ structure,...

    2023

  47. [47]

    Chan, Phys

    L.-H. Chan, Phys. Rev. Lett. 54, 1222 (1985); ibid. 55, 2 1 (1985)

    1985

  48. [48]

    K. S. Babu, A. K. Das and P. Panigrahi, Phys. Rev. D 36 (1987), 3725

    1987

  49. [49]

    G. V. Efimov, Nonlocal Interactions of Quantized Fields , (Nauka, 1993)

    1993

  50. [50]

    Talaganis, T

    S. Talaganis, T. Biswas and A. Mazumdar, Class. Quant. G rav. 32, 215017 (2015) [arXiv: 1412.3467 [hep-th]]

    Pith/arXiv arXiv 2015

  51. [51]

    Buoninfante, G

    L. Buoninfante, G. Lambiase, Y. Miyashita, W. Takebe an d M. Yamaguchi, Phys. Rev. D 101 (2020) no.8, 084019 [arXiv:2001.07830 [hep-th]]

    Pith/arXiv arXiv 2020

  52. [52]

    G. H. Hardy, Divergent Series, (Oxford University Pres s, 1949)

    1949

  53. [53]

    E. T. Copson, Asyptotic Expansions, (Cambridge Univer sity Press, 1965)

    1965

  54. [54]

    A. A. Saharian, Generalized Abel-Plana Formula as a Renormalization Tool in Quantum Field Theory, [arXiv: 2409.05465 [hep-th]]. 19

    Pith/arXiv arXiv