Superrenormalizable gauge and gravitational theories

E. T. Tomboulis (UCLA)

REVIEW

Superrenormalizable gauge and gravitational theories

E. T. Tomboulis (UCLA)

High Energy Physics - Theory · Submitted 20 Feb 1997 · 11 references

Pith's verdict: An explicit, checkable construction of a superrenormalizable nonlocal gauge/gravity theory; the perturbative story holds together, the nonperturbative one is left open and the paper says so.

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arXiv IDhep-th/9702146 (version 1)

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CitationE. T. Tomboulis (UCLA). "Superrenormalizable gauge and gravitational theories." arXiv:hep-th/9702146 (1997). https://pith.science/paper/INNOK2IB

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Editor's note

read the letter

Worth your time if you care about nonlocal UV completions, but read it as a construction paper, not a proof of a QFT. The core is real: §4 gives an explicit h(z) = exp(H(z)) with H built from a decaying entire ζ, and the no-pole condition is achieved by choosing the renormalization scale so that 1 + g²α h = exp(H), which never vanishes. That is a clean trick, and the argument that no polynomial can do both jobs is correct and worth having stated. Appendix A's power counting is the strongest part of the paper — detailed, case-by-case, and it actually delivers δ_G = 4 − 2γ(L−1) − E. The generalized KL representation in §7 is a genuinely useful consistency check: it shows the improved UV behavior comes entirely from the bare h⁻¹ factors and not from self-energy insertions, which is exactly what you want to see if the theory is to stay renormalizable without h being renormalized. That section is the paper's best evidence that the pieces fit together rather than merely coexist. On the soft spots, I'd push back on the reader's emphasis. Appendix C's term-by-term Veltman argument is asserted, yes, but the paper supplies the ingredients (absolute convergence, quasi-locality, the propagator decomposition) and this is the rigor level of the era. The sharper issue is the one the stress-test names: h grows doubly-exponentially in the even sectors, the whole analysis is Euclidean-then-continued, and §9 concedes that global stability is unexamined. That concession is honest, but it is the question that decides whether this is a theory or a formal series, and it gets one sentence. Second, smaller: the claim in §4 that the spurious zeros at other scales cancel against self-energy is argued via (4.8) at the level of the inverse propagator, not demonstrated for the resummed propagator. Real gap, not fatal. The §9 talk of string-like bound states is speculation and reads as such. Send it out. It deserves referees who will check Appendix A line by line and press on the Euclidean continuation.

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Major objections (3)

[§4, Eq. (4.8)] The RG-invariance argument for the cancellation of the infinitely many complex poles at scales µ ≠ µ0 is schematic. Eq. (4.8) is written at the level of the inverse 2-point function's functional form, but the actual cancellation of the pole contributions in the dressed propagator is not demonstrated. Since the cutting rules and unitarity proof are performed at µ0, the paper should either explicitly restrict the unitarity/causality claims to µ0 or give a more concrete demonstration that the pole contributions cancel. This is load-bearing for claim (ii) 'no unphysical poles': the reader needs to know whether this holds only at a special scale or generally.
[Appendix C] The term-by-term application of the Veltman largest-time equation to the infinite series of quasi-local vertices is asserted ('By standard theorems...') rather than proved. The paper names the ingredients (absolute convergence, quasi-locality, propagator decomposition (6.6)), but does not specify the topology or test-function space on which the series converges uniformly. Since the cutting equations (abstract:s3) are a headline result, a sentence making the convergence statement explicit would strengthen the paper.
[§5 and §9] The doubly-exponential growth of h(z) in the even-numbered sectors (4.10) means the Euclidean action is not obviously bounded below. The paper acknowledges this in §9 ('Harder to investigate issues outside perturbation theory, such as global stability, may lead to further constraints'). Since the claims are explicitly perturbative, this is an acknowledged scope limitation, not a flaw in the stated claims; but it is worth stating clearly that the construction establishes perturbative superrenormalizability, not the existence of the theory as a QFT.

Minor objections (10)

[Eq. (2.3) and Eq. (4.1)] There is a notational inconsistency between h and h. Eq. (2.3) writes the denominator as 1 + g²αh(k²/Λ²), but Eq. (4.1) defines h ≡ 1 + g²αh. The propagator should be written with h. This is confusing.
[Eq. (4.11)] The bound has a stray bracket and the factor '(1-ε)ε)' appears to be a typo for '(1-ε)'. Please check.
[Eq. (8.13)] The apparent 1/z singularity at z = 0 is cancelled by the vanishing of e^H(z) − 1 at z = 0 (since H(0) = 0). A sentence noting this would help.
[§7, Eq. (7.12)] The extension to x⁰ = y⁰ is asserted via Appendix C; a sentence of detail would help.
[§9] The 'extended bound excitations' and 'string-like excitations' are speculative and not backed by calculation. The paper says 'can be expected' and 'it is plausible.' This is fine as speculation but should be flagged as such.
[§9] The claim that the abelian version coupled to fermions is 'a version of QED with smooth UV behavior and apparently no UV Landau pole' is speculative and not backed by calculation. Same for the supersymmetrization claim.
[§2] The paper says 'the FP ghost term... does not depend on w.' This is correct, but the reader might wonder why the ghost propagator is unchanged. A sentence would help.
[§3, Eq. (3.3)] The combinatorial factor Γ = l!²/(2l−n)! is not derived in the text; a brief derivation or reference would help.
[§6] The paper says 'the vertices V̂, eq. (6.8), are real for real momenta.' This relies on h being real and positive on the real axis (condition (i)). A sentence making this explicit would help.
[§8, Eq. (8.10)] The paper says 'if γ ≥ 3, only 1-loop diagrams are superficially divergent.' But Eq. (8.10) gives δG ≤ 4 − 2γ(L−1), so for γ = 2 and L = 2, δG ≤ 0 (log divergent). The paper's requirement γ ≥ 3 is correct for killing 2-loop divergences, but the text should note that γ = 2 would leave 2-loop log divergences.

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The paper

The paper claims

A gauge theory can be made superrenormalizable without ghosts, if the extra derivatives are summed into a carefully chosen entire function rather than a polynomial.

The reading

Adding higher-derivative terms to a gauge action is the classic way to tame ultraviolet divergences, but any finite polynomial of the covariant d'Alembertian leaves the transverse propagator with extra poles whose residues must be negative — ghosts. This paper asks whether that trade-off is forced, and argues it is not: if the extra term is an infinite series representing a transcendental entire function $h(z)$ with no zeros anywhere in the finite complex plane, the denominator $1+g^2a h(z)$ can be written as $e^{H(z)}$, which never vanishes, so no new poles appear. The author constructs an explicit class of such functions and shows by detailed power counting that only one-loop divergences with two, three, or four external gauge legs survive, all gauge invariant and removable by a single $F^2$ counterterm. Cutting equations, a generalized Källén-Lehmann representation, and a version of the same construction for gravity are then presented to argue that unitarity, causality, and positivity survive the modification within the loop expansion.

What carries the argument

The load-bearing object is the pair $H(z) = ∫_0^{p_γ(z)} (1-ζ(w))/w dw$ and $h(z) = e^{H(z)} - 1$, with $p_γ$ a real polynomial of degree $γ$ vanishing at the origin and $ζ$ an entire function equal to one at the origin that decays to zero in specified angular sectors — for instance $ζ(z) = exp(-(C_n z^{2n} + ... + C_1 z^2))$. Because $ζ$ decays in alternating sectors of angle $π/4n$ and blows up in the others, $H$ behaves logarithmically (so $h$ behaves polynomially, like $|z|^γ$) in cones around the real axis, while $h$ grows doubly exponentially elsewhere; the entire-function growth theory invoked is what makes this directional behaviour possible. A second piece of machinery is the rewriting $D^{μν}(k) = h^{-1/2}(k^2) D^{μν}(k) h^{-1/2}(k^2)$, which pushes the $h$ factors off the propagator and onto the vertices and sources. That restores the naive equal-time decomposition of the propagator, so the Veltman largest-time argument and the resulting Cutkosky-style cutting rules can be run graph by graph, and the Bogoliubov causality condition follows.

Extended reading notes

Core claim

The claim is that a four-dimensional gauge theory with the Lagrangian $-F^2/2g^2 - (a/2)F h(-D^2/Λ^2)F$ is perturbatively superrenormalizable and free of gauge-invariant unphysical poles, provided $h$ is a transcendental entire function that is real and positive on the real axis, has no zeros anywhere, and grows at most like $|z|^γ$ with $γ ≥ 2$ in cones around the real axis. The obstruction for polynomials is stated sharply: partial fractions applied to $1/(k^2(1+g^2a p_n(k^2)))$ forces at least one negative residue, so higher-derivative regularization always pays for its improved convergence with ghosts. Transcendental $h$ escapes this because $1+g^2a h(z)$ can be made equal to $e^{H(z)}$, a zero-free function, and the coupling can be renormalized at a scale where that identity holds. Power counting on the resulting infinite tower of vertices then gives a superficial degree of divergence $δ_G = 4 - 2γ(L-1) - E$, so multi-loop graphs and one-loop graphs with more than four external legs are convergent, and $h$ itself is never renormalized. The paper extends the same construction to fourth-order gravity with two entire functions $h_2$ and $h_0$, where $γ ≥ 3$ leaves only one-loop gauge-invariant divergences.

Load-bearing premise

The unitarity and causality conclusions depend on the assertion, in Appendix C, that the Veltman largest-time argument can be applied term by term to the infinite series of quasi-local vertices and that the Fourier transforms of those vertices converge as distributions — a step justified only by a reference to standard theorems rather than carried out.

Editorial extensions

If this is right

Ultraviolet behaviour of gauge and gravitational theories can be improved by an infinite derivative series without the negative-residue poles that any finite polynomial series forces, so the standard no-go argument against ghost-free higher-derivative regularization is evaded rather than circumvented.
Only one-loop divergences with at most four external gauge legs survive, and they are gauge invariant, so renormalization needs just the $F^2$ counterterm and the function $h$ is protected from renormalization.
The generalized Källén-Lehmann representation shows the spectral density stays positive while the propagator falls faster than $1/p^2$, so improved convergence and physical positivity are compatible rather than in tension.
At distances shorter than $1/Λ$ the inverse propagator is a polynomial of degree $2+2γ$ (or $4+2γ$ in gravity), giving a steep short-distance confining potential and a spectrum of tightly bound extended excitations that the author suggests could be relevant to gravitational collapse and spacetime singularities.
Minimally coupled fermions leave the superrenormalizability intact, whereas self-interacting scalars reintroduce multi-loop divergences through matter subgraphs.

Reading between the lines

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

The zero-free condition on $1+g^2a h(z)$ is really a condition on the renormalization point: at any other scale the bare propagator does acquire infinitely many complex poles, and the paper's argument that they cancel against self-energy corrections is a renormalization-group consistency statement that deserves an explicit two-loop check.
If the construction holds, the natural next test is whether the one-loop finiteness of the would-be divergent structures survives when the entire function is truncated at increasing order $N$ — a sequence of polynomial theories whose ghosts should decouple as $N → ∞$ would be a concrete way to see the mechanism at work.
The claim that the short-distance potential is confining at tree level suggests a calculational programme: bound-state spectra of gauge bosons in this theory could be computed in the nonrelativistic limit and compared with the string-like Regge behaviour the author speculates about.
Because the modified vertices are entire functions of momenta, the theory has no Landau pole in the abelian case; whether the same is true non-abelian, where the one-loop $F^2$ counterterm still runs, is left open by the paper and is a sharp question for the programme.

Original abstract

We investigate 4-dim gauge theories and gravitational theories with nonpolynomial actions containing an infinite series in covariant derivatives of the fields representing the expansion of a transcendental entire function. A class of entire functions is explicitly constructed such that: (i) the theory is perturbatively superrenormalizable; (ii) no (gauge-invariant) unphysical poles are introduced in the propagators. The nonpolynomial nature is essential; it is not possible to simultaneously satisfy (i) and (ii) with any polynomial series in derivatives. Cutting equations are derived verifying the absence of unphysical cuts and the Bogoliubov causality condition within the loop expansion. A generalized KL representation for the 2-point function is obtained exhibiting the consistency of physical positivity with the improved convergence of the propagators. Some physical effects, such as extended bound excitations in the spectrum, are briefly discussed.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We investigate 4-dim gauge theories and gravitational theories with nonpolynomial actions containing an infinite series in covariant derivatives of the fields representing the expansion of a transcendental entire function. A class of entire functions is explicitly constructed such that: (i) the theory is perturbatively superrenormalizable; (ii) no (gauge-invariant) unphysical poles are introduced in the propagators. The nonpolynomial nature is essential; it is not possible to simultaneously satisfy (i) and (ii) with any polynomial series in derivatives. Cutting equations are derived verifying the absence of unphysical cuts and the Bogoliubov causality condition within the loop expansion. A generalized KL representation for the 2-point function is obtained exhibiting the consistency of physical positivity with the improved convergence of the propagators. Some physical effects, such as extended bound excitations in the spectrum, are briefly discussed.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate 4-dim gauge theories and gravitational theories with nonpolynomial actions containing an infinite series in covariant derivatives of the fields representing the expansion of a transcendental entire function.	contribution 0.85	0.80	0.05	asserted 0.91
s1	A class of entire functions is explicitly constructed such that: (i) the theory is perturbatively superrenormalizable; (ii) no (gauge-invariant) unphysical poles are introduced in the propagators.	contribution 0.57	0.95	0.53	asserted 1.00
s2	The nonpolynomial nature is essential; it is not possible to simultaneously satisfy (i) and (ii) with any polynomial series in derivatives.	result 0.88	0.78	0.45	asserted 1.00
s3	Cutting equations are derived verifying the absence of unphysical cuts and the Bogoliubov causality condition within the loop expansion.	result 0.65	0.95	0.09	asserted 1.00
s4	A generalized KL representation for the 2-point function is obtained exhibiting the consistency of physical positivity with the improved convergence of the propagators.	result 0.81	0.94	0.11	asserted 0.91
s5	Some physical effects, such as extended bound excitations in the spectrum, are briefly discussed.	interpretation 0.68	0.71	0.04	asserted 0.57

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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full text
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References

Works cited (11)

[1]

A. A. Slavnov, Theor. Math. Phys. 13 (1972) 174; 33 (1977) 977

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[2]

T. D. Bakeyev and A. A. Slavnov, preprint SMI-02-96, hep- th/9601092

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[3]

G. V. Eﬁmov, Commun. Math. Phys. 5 (1967) 42; V. A. Alebastrov and G. V. Eﬁmov, Commun. Math. Phys. 38 (1974) 11

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[4]

M. Kato, Phys. Lett. 245B (1990) 43

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[5]

E. C. Titchmarsh, The theory of Functions (Oxford University Press, 2nd corrected edition, 1968); B. Ja. Levin, Distribution of zeros of entire functions (American Mathematical Society, Providence, Rhode Island, 1964)

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[6]

S. Weinberg, Phys. Rev. 118, 838 (1960)

View on Pith 1960
[7]

W. Zimmermann, Comm. Math. Phys. 11 (1968) 1; Y. Hahn and W. Zimmermann, Comm. Math. Phys. 10 (1968) 330

View on Pith 1968
[8]

M. Veltman, Physica 29 (1963) 186; G. ’t Hooft and M. Veltman, Diagrammar in Particle interactions at very high energies , D. Speiser, F. Halzen and J. Weyers, eds. (Plenum Press London 1974)

View on Pith 1963
[9]

G. ’t Hooft, Nucl. Phys. B33 (1971) 173; G. ’t Hooft and M. Veltman, Nucl. Phys. B50 (1972) 318

View on Pith 1971
[10]

N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Wiley-Interscience, New York, 1959)

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[11]

E. Remiddi, Helvetica Phys. Acta, 54, 365 (1981). 40

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Counting ghosts in the "ghost-free" non-local gravity
hep-th 2015-01 conditional novelty 5.0 of 10

The exponential non-local gravity that avoids massive ghosts at tree level is argued to acquire infinitely many complex ghost poles from quantum corrections.
On Newtonian singularities in higher derivative gravity models
hep-th 2014-12 conditional novelty 5.0 of 10

For local higher-derivative gravity with real simple propagator poles, the point-mass Newtonian potential is finite at the origin because massive tensor and scalar mode contributions cancel the 1/r term.
Super-renormalizable & Finite Gravitational Theories
hep-th 2014-07 conditional novelty 5.0 of 10

A non-polynomial higher-derivative gravity theory can be made one-loop finite in four dimensions by adding two quartic curvature operators whose coefficients cancel the R-squared and Ricci-squared beta functions.
Super-renormalizable Quantum Gravity
hep-th 2011-07 conditional novelty 5.0 of 10

An infinite-derivative extension of Stelle gravity is argued to be ghost-free and super-renormalizable, with a spectral dimension below one in the ultraviolet and singularity-free black holes.
On the Meaning of Localization in Non-Local Quantum Field Theory
physics.gen-ph 2026-06 unverdicted novelty 4.0 of 10

The paper derives a nonlocal phase-space uncertainty relation implying a minimal measurable length of order L_M and a finite phase-space cell in nonlocal QFT.
Emerging Nonlocal K\"{a}ll\`{e}n-Lehmann Higgs Spectra at the LHC
hep-ph 2026-05 unverdicted novelty 4.0 of 10

Nonlocal Källén-Lehmann spectral densities in the Higgs sector yield exponentially suppressed scattering amplitudes above Λ_NL and suppress the real part of the Higgs self-energy at p² ~ -Λ²_NL, solving the hierarchy ...
On Gauge-Invariant Entire-Function Regulators and UV Finiteness in NonLocal Quantum Field Theory
hep-th 2025-11 unverdicted novelty 4.0 of 10

In flat space, an entire-function regulator F(□/M^2) acts as the multiplicative Euclidean form factor e^{-p_E^2/M^2} on plane waves, yielding exponential UV damping; this known nonlocal-QFT result is re-derived and discussed.
On Recent measurements of Toponium Threshold Enhancement in Entire-Function-Regulated Nonlocal Quantum Field Theory
physics.gen-ph 2025-07 unverdicted novelty 4.0 of 10

Threshold excess in toponium production is accommodated in an entire-function-regulated nonlocal QFT by a data-driven cutoff parameter and small RG effects while preserving global QCD tests.
Spacetime completeness of non-singular black holes in conformal gravity
gr-qc 2016-11 conditional novelty 4.0 of 10

By rescaling the Schwarzschild and Kerr metrics with a conformal factor that diverges at the singularity, the authors obtain regular, geodesically complete black hole spacetimes in conformal gravity.
Behavior of the Newtonian potential for ghost-free gravity and singularity-free gravity
gr-qc 2016-04 conditional novelty 4.0 of 10

Infinite-derivative ghost-free gravity universally predicts a non-singular Newtonian potential that reduces to 1/r in the infrared, with table-top experiments bounding the non-locality scale above 0.004 eV.
Quantum scale symmetry
hep-th 2019-01 conditional novelty 2.5 of 10

Scale symmetry in quantum field theory arises at fixed points, and its approximate breaking near those fixed points may link the Higgs hierarchy, quantum gravity, and cosmic acceleration.
De Sitter Cores from Nonlocal Quantum Field Theories
physics.gen-ph 2026-07 conditional novelty 2.0 of 10

An exponential nonlocal regulator maps a point mass to a Gaussian density whose Einstein solution has a de Sitter core, reproducing (with new labeling) the known Gaussian-sourced regular black hole.
Quantum gravity phenomenology at the dawn of the multi-messenger era -- A review
hep-ph 2021-11 accept novelty 2.0 of 10

The review argues that multi-messenger astronomy now supplies workable empirical constraints on Planck-scale quantum gravity, chiefly through energy-dependent propagation effects accumulated over cosmological distances.

How later papers used it

135 later papers cite it

Evidence: substantial. 134 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 11 of the 214 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 1 built on it and 1 used its method or tool.

Builds on it 1 of 2 papers with this label

Proposed a generalization of the form factor to this paper's form, using it to obtain predictions for the tensor-to-scalar ratio as a function of the non-locality scale.

Non-local Starobinsky inflation in the light of future CMB (2018) · 0.90 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 paper

Added gauge-fixing terms according to this paper.

The Nonsymmetric Kaluza-Klein Theory and Modern Physics. A Novel Approach (2014) · 0.93 · paraphrased: the licence does not allow quoting

How the literature uses it

140 arXiv papers cite this in 222 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 130 sentences without a confident label

Builds on it 2 papers

We also propose a generalization of the form factor to [ref] in ([ref]) using which we obtain the predictions $5×10^-4< r≲0.07$ as a function of non-locality scale $5×10^-5 </M_p< 10^-4$.

Non-local Starobinsky inflation in the light of future CMB · Conclusions · 0.90
We extend the finite UV-complete nonlocal QFT framework [ref] to the heavy-quark sector and derive the modified Bethe–Salpeter (BS) equation for a color-singlet $tt̅$ pair.

On Recent measurements of Toponium Threshold Enhancement in Entire-Function-Regulated Nonlocal Quantum Field Theory · Nonlocal Bethe–Salpeter Theory for Toponium · 0.97

Uses its method or tool 1 paper

The construction of such s can be done according to Refs [ref] .

The Nonsymmetric Kaluza-Klein Theory and Modern Physics. A Novel Approach · Conclusions and prospects for further research · 0.93

Compares with it 1 paper

We confirm that in the limit of large $N$, we recover known results from the literature on nonlocal gravity [ref] .

Asymptotically nonlocal gravity · Examples of emergent scale · 0.82

Cites it as support 3 papers

In fact, renormalizability seems to be a feature of higher derivative theories [ref] , due to the fact that the presence of higher order derivatives implies that the graviton propagator goes to zero more rapidly than $1/k^2$ at high wave numbers $k$.

On the true nature of renormalizability in Horava-Lifshitz gravity · Introduction · 0.99
However, it has been shown that for a specific class of non-local theories we are interested in here, the violation of causality is limited to the scale of non-locality [ref] . ].

Locality & Entanglement in Table-Top Testing of the Quantum Nature of Linearized Gravity · Underlying Assumptions · 0.98
The difference comes from the fact that in the latter case the exponent is decaying along both real and complex axis and necessity of this was noted already in [ref] .

Non-local self-healing of Higgs inflation · Introduction · 0.96

Cites it as background 72 papers

A substantial literature exists on higher-derivative theories, including those with quadratic term that are modified by an operator involving finite or infinite numbers of derivatives [ref] .

Asymptotic nonlocality in non-Abelian gauge theories · Introduction · 0.99
Quantum field theories involving higher-derivative quadratic terms have been of substantial interest due to the improvement in the short-distance behavior of amplitudes [ref] .

Asymptotic nonlocality in gauge theories · Introduction · 0.97
One can go further and consider a gravity theory with infinite derivative operators often named analytic infinite derivative gravity [ref] .

Non-perturbative quantum gravity denounces singular Black Holes · Introduction · 0.83

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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