Strong Interactions and Stability in the DGP Model

Markus A. Luty; Massimo Porrati; Riccardo Rattazzi

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Strong Interactions and Stability in the DGP Model

Markus A. Luty, Massimo Porrati, Riccardo Rattazzi

High Energy Physics - Theory · Submitted 12 Mar 2003 · 16 references

Pith's verdict: The paper that actually computes the DGP strong-coupling scale rather than guessing it, with one printed formula that doesn't reproduce its own result.

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arXiv IDhep-th/0303116 (version 1)

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CitationMarkus A. Luty, Massimo Porrati, Riccardo Rattazzi. "Strong Interactions and Stability in the DGP Model." arXiv:hep-th/0303116 (2003). https://pith.science/paper/RCPR2BDG

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Editor's note

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Bottom line: this is the paper that pins down where DGP stops being predictive, and it does it by computing rather than by analogy. The λ3 ~ (m²M4)^{1/3} claim in the abstract is earned in Sections 2 and 3, not asserted, and that is the difference between this and the earlier hand-waving about running the massive-gravity cutoff into |p|/λDGP (which the authors themselves flag in Section 1 as a heuristic). Worth your time if you care about DGP, massive gravity, or modified-gravity model building generally. The 1000 km number is the one everybody quotes, and it holds up here. What is genuinely new is the machinery: the boundary effective action with free boundary conditions (Section 2, Section 3), the identification of the strongly coupled mode as a brane bending that is pure gauge in the bulk, has H_μν = 0, and obeys de Donder (Eq. 3.26), the diagonalization that shows π gets its kinetic term only through mixing (Eq. 3.24), and the explicit cubic vertex M4² m ∂π∂π□π in Eq. 3.38. The non-renormalization check at Eq. 3.42 is a nice touch and it is correct. Section 4 is a clean Birkhoff-plus-junction computation, and the observation that the negative-energy solution sits at curvature of order m², exactly where the derivative expansion dies, is a real consistency argument rather than rhetoric. Soft spots, in order of size. First, Eq. 3.30 does not reproduce Eq. 3.31: put p = 0, q = 3 into the printed exponent and you get (m²M4)^{1/7}, not ^{1/3}. That is a typo in the general power-counting formula, not in the physics, since Eq. 3.38 fixes the cubic coefficient directly. Still, it is the formula people will copy, so it deserves an erratum. Second, Section 5 is the thinnest part. Eq. 5.22 leans on the small-wavelength replacement Eq. 5.13, justified by a flat-space limit and checked explicitly only for de Sitter. The RS IR-brane ghost claim is cross-checked against the radion kinetic term Eq. 5.43, which is reassuring, but the general curved case is sketched. On the tuning worry in your report: I don't think it lands. The cubic is tree level from the bulk Einstein action, canonical normalization is fixed by Eq. 3.24, so the coupling is 1/Λ³ with no freedom to dial. The massive-gravity analogy cuts the other way, since there Λ3 is the floor. Send it out. This is a serious paper and the referees should engage with it.

Referee report

Minor objections (7)

[Section 3, Eq. (3.30)] Evaluating the displayed power-counting formula at p=0, q=3 gives (m^2 M4)^{1/7}, whereas Eq. (3.31) and the explicit computation in Eq. (3.38) give (m^2 M4)^{1/3}. The discrepancy arises because the schematic derivative counting in Eq. (3.28) assigns one derivative per field, while the leading cubic term ∂π∂π□π carries four derivatives; the formula should be corrected or the counting convention stated explicitly.
[Abstract] The phrase 'at the same scale scale λ3' duplicates the word 'scale' and should read 'at the same scale λ3'.
[Section 1] The sentence 'Because the scale R* is very large for astrophysical sources, it appeas that the DGP model may describe our universe [2]' contains a typo: 'appeas' should be 'appears'. The same paragraph also has 'This property suggest' where 'suggests' is intended.
[Section 5.1] The word 'Minkowsky' should be 'Minkowski'.
[Section 5.3] The estimate that the strong-coupling scale on the Planck brane is (M5 k)^{1/2} is used to support the holographic interpretation but is not derived; a short derivation or a reference would make this step checkable.
[Section 3, Eq. (3.28)] The notation ∂(N)^p(∂π)^q is ambiguous about whether the derivatives act on individual fields or on the product; please define the convention, since it feeds directly into the power-counting formula of Eq. (3.30).
[Reference [8]] Reference [8] is cited only as arXiv:hep-th/0210184; if it was unpublished at the time of writing, that should be noted so readers can judge the status of the massive-gravity comparison.

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The paper

The paper claims

The DGP braneworld is strongly interacting at a length scale $�lambda_3 �sim (�lambda_{�rm DGP}^2/M_4)^{1/3}$, about 1000 km for a Hubble-scale crossover.

The reading

The paper asks whether the DGP braneworld—a 5D bulk with a 4D Einstein term on a brane—really behaves as a weakly coupled 4D gravity theory at distances shorter than its crossover scale $�lambda_{�rm DGP}$. It argues that it does not: a longitudinal brane-bending mode, the Goldstone field eaten by the graviton, has no independent kinetic term at leading order and gets one only by mixing with transverse graviton polarizations. That small kinetic term makes the mode strongly interacting at $�lambda_3 �sim (�lambda_{�rm DGP}^2/M_4)^{1/3}$, which is about 1000 km if $�lambda_{�rm DGP}$ is of order the Hubble length. The paper also finds negative-energy classical solutions at the edge of the effective theory and shows that positive curvature can turn the Goldstone mode into a ghost. The stake is whether DGP can be a predictive alternative to dark energy, or whether it requires new physics at a short distance that is still far larger than the Planck length.

What carries the argument

The argument rests on a boundary effective action for 5D gravity with no a priori boundary condition. Splitting the metric into lapse, shift, and 4D metric, fixing de Donder gauge in the bulk, and solving the bulk equations with fixed boundary values produces a boundary action whose residual gauge freedom is parameterized by a brane-bending shift $�Xi^5=e^{-y�Delta}�pi$. This mode is pure gauge in the bulk, leaves the induced 4D metric unchanged at linear order, and therefore has no large DGP kinetic term; its kinetic term arises only from mixing with the transverse graviton, which is the technical origin of the strong coupling. The same formalism yields the cubic interaction and, on curved backgrounds, the curvature-dependent $�pi$ kinetic term that controls the ghost. A separate 5D Schwarzschild solution with an $S^3$ boundary supplies the negative-energy example and shows that the instability sits at curvature $O(m^2)$.

Extended reading notes

Core claim

The central claim is that DGP has a hidden strong-coupling scale $�lambda_3 �sim (�lambda_{�rm DGP}^2/M_4)^{1/3}$, not just the crossover scale $�lambda_{�rm DGP}$ at which gravity becomes five-dimensional. The paper derives this by integrating out the bulk with free boundary conditions and organizing the boundary action around the brane-bending mode $�pi$; at quadratic order $�pi$ has kinetic term $3M_4^2 m^2 �pi�,�Box�pi$ (Eq. (3.24)), suppressed by $m=M_5^3/M_4^2$, and at cubic order the explicit interaction $M_4^2 m�,�partial_�mu�pi�,�partial^�mu�pi�,�Box�pi$ (Eq. (3.38)) makes the mode strongly coupled at $�Lambda �sim (m^2M_4)^{1/3}$. For $�lambda_{�rm DGP}$ of order the Hubble length this gives $�lambda_3 �sim 1000$ km, below which the effective theory loses predictivity. The paper adds a negative-energy classical solution satisfying the dominant energy condition, confined to curvature of order $m^2$, and shows that in de Sitter-like backgrounds the $�pi$ kinetic term becomes $3M_4^2 m(m-2C)�pi�,�Box�pi$, so the mode is a ghost when the background curvature $C$ reaches $m/2$.

Load-bearing premise

The derivation assumes that every boundary operator allowed by 4D general covariance enters with a coefficient of order one at the cutoff, so that no tuned counterterm raises the strong-coupling scale or gives the Goldstone mode a large independent kinetic term.

Editorial extensions

If this is right

If $�lambda_{�rm DGP}$ is of order the Hubble length, DGP predictions for gravity at distances below about 1000 km require a UV completion; the model cannot be treated as a complete 4D effective theory there.
The Vainshtein recovery of general relativity around a source at $R_*�sim(R_S�lambda_{�rm DGP}^2)^{1/3}$ occurs where the 4D curvature is $O(m^2)$, so the recovery is not established within the quantum effective theory that the paper analyzes.
A cutoff at $�lambda_3$ removes the negative-energy classical solutions, because those solutions have $|R_S|<1/m$ and curvature $O(m^2)$.
In a de Sitter-like background the Goldstone mode becomes a ghost when $C=m/2$, so the self-accelerating DGP background is unstable in this effective description.
A DGP kinetic term on the Planck brane of Randall-Sundrum does not introduce a new strong-coupling scale below the AdS scale, while a DGP term on the IR brane makes the radion a ghost when $m<2k$.

Reading between the lines

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

The paper's power counting treats all allowed boundary operators as entering at the cutoff; a natural next step is to ask whether DGP admits a tuned counterterm analogous to the massive-gravity tuning that raises $�Lambda_5$ to $�Lambda_3$, which would weaken the 1000 km claim to a statement about a particular matching.
Because the ghost appears for positive curvature, a consistent UV completion of DGP self-acceleration must either remove the ghost or forbid the background; this links the model's cosmological use to the same curved-space stability questions that arise in massive gravity.
The negative-energy solutions suggest a general relationship between dominant-energy brane matter and negative 5D mass in braneworlds; a scan of the allowed equation-of-state and mass range could show whether the $�lambda_3$ cutoff is necessary or merely sufficient to exclude them.

Original abstract

The model of Dvali, Gabadadze, and Porrati (DGP) gives a simple geometrical setup in which gravity becomes 5-dimensional at distances larger than a length scale \lambda_{DGP}. We show that this theory has strong interactions at a length scale \lambda_3 ~ (\lambda_{DGP}^2 / M_P)^{1/3}. If \lambda_{DGP} is of order the Hubble length, then the theory loses predictivity at distances shorter than \lambda_3 ~ 1000 km. The strong interaction can be viewed as arising from a longitudinal `eaten Goldstone' mode that gets a small kinetic term only from mixing with transverse graviton polarizations, analogous to the case of massive gravity. We also present a negative-energy classical solution, which can be avoided by cutting off the theory at the same scale scale \lambda_3. Finally, we examine the dynamics of the longitudinal Goldstone mode when the background geometry is curved.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,004 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.81 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.31 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

The model of Dvali, Gabadadze, and Porrati (DGP) gives a simple geometrical setup in which gravity becomes 5-dimensional at distances larger than a length scale \lambda_{DGP}. We show that this theory has strong interactions at a length scale \lambda_3 ~ (\lambda_{DGP}^2 / M_P)^{1/3}. If \lambda_{DGP} is of order the Hubble length, then the theory loses predictivity at distances shorter than \lambda_3 ~ 1000 km. The strong interaction can be viewed as arising from a longitudinal `eaten Goldstone' mode that gets a small kinetic term only from mixing with transverse graviton polarizations, analogous to the case of massive gravity. We also present a negative-energy classical solution, which can be avoided by cutting off the theory at the same scale scale \lambda_3. Finally, we examine the dynamics of the longitudinal Goldstone mode when the background geometry is curved.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The model of Dvali, Gabadadze, and Porrati (DGP) gives a simple geometrical setup in which gravity becomes 5-dimensional at distances larger than a length scale \lambda_{DGP}.	background 1.00	0.06	0.04	asserted 1.00
s1	We show that this theory has strong interactions at a length scale \lambda_3 ~ (\lambda_{DGP}^2 / M_P)^{1/3}.	result 0.98	0.96	0.66	asserted 1.00
s2	If \lambda_{DGP} is of order the Hubble length, then the theory loses predictivity at distances shorter than \lambda_3 ~ 1000 km.	result 0.59	0.80	0.31	asserted 0.94
s3	The strong interaction can be viewed as arising from a longitudinal `eaten Goldstone' mode that gets a small kinetic term only from mixing with transverse graviton polarizations, analogous to the case of massive gravity.	interpretation 1.00	0.70	0.11	asserted 0.68
s4	We also present a negative-energy classical solution, which can be avoided by cutting off the theory at the same scale scale \lambda_3.	result 0.93	0.94	0.13	asserted 0.95
s5	Finally, we examine the dynamics of the longitudinal Goldstone mode when the background geometry is curved.	contribution 0.99	0.87	0.04	unclear 0.75

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
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references and citing sentences
relations to other papers

levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (16)

[1]

G. R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. B 485, 208 (2000) [arXiv:hep-th/0005016]

View on Pith Pith review arXiv 2000
[2]

C. Deﬀayet, G. R. Dvali and G. Gabadadze, Phys. Rev. D 65, 044023 (2002) [arXiv:astro-ph/0105068]

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[3]

H. van Dam and M. J. Veltman, Nucl. Phys. B 22, 397 (1970); V. I. Zakharov, JETP Lett. 12, 312 (1970)

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[4]

A. I. Vainshtein, Phys. Lett. B 39, 393 (1972)

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[5]

C. Deﬀayet, G. R. Dvali, G. Gabadadze and A. I. Vainshtein, Phys . Rev. D 65, 044026 (2002) [arXiv:hep-th/0106001]

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[6]

A. Gruzinov, arXiv:astro-ph/0112246

View on Pith Pith review arXiv
[7]

M. Porrati, Phys. Lett. B 534, 209 (2002) [arXiv:hep-th/0203014]

View on Pith Pith review arXiv 2002
[8]

N. Arkani-Hamed, H. Georgi and M. D. Schwartz, arXiv:hep-th/ 0210184

View on Pith
[9]

M. Porrati, Phys. Lett. B 498, 92 (2001) [arXiv:hep-th/0011152]; I. I. Ko- gan, S. Mouslopoulos and A. Papazoglou, Phys. Lett. B 503, 173 (2001) [arXiv:hep-th/0011138]

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[10]

L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999) [arXiv:hep-ph/9905221]; L. Randall and R. Sundrum, Phys. Rev. Le tt. 83, 4690 (1999) [arXiv:hep-th/9906064]. 23

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[11]

R. Arnowitt, S. Deser and C. Misner, in Gravitation: An Introduction to Current Research, ed. L. Witten (New York, Wiley) 1962

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G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977); S. W. Hawking and G. T. Horowitz, Class. Quant. Grav. 13, 1487 (1996) [arXiv:gr-qc/9501014]

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[15]

M. Porrati, JHEP 0204, 058 (2002) [arXiv:hep-th/0112166]

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[16]

C. Charmousis, R. Gregory and V. A. Rubakov, Phys. Rev. D 62, 067505 (2000) [arXiv:hep-th/9912160]. 24

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Massive Galileon Positivity Bounds
hep-th 2017-02 accept novelty 6.0 of 10

A massive Galileon can satisfy the positivity bounds required for a local, analytic ultraviolet completion, and the authors construct an explicit renormalizable example.
Chameleon Dark Energy and Atom Interferometry
astro-ph.CO 2016-03 conditional novelty 6.0 of 10

A full 3D solve of the chameleon field in the Berkeley atom-interferometry chamber confirms the thin-shell approximation to within 20% and leaves the chameleon exclusion bound unchanged.
Positive Signs in Massive Gravity
hep-th 2016-01 conditional novelty 6.0 of 10

Positivity of forward massive-graviton scattering amplitudes excludes most of the (c3, d5) parameter space of ghost-free massive gravity, leaving a compact allowed island.
Counting the degrees of freedom of generalized Galileons
gr-qc 2015-06 conditional novelty 6.0 of 10

The third-order time derivatives in generalized Galileon field equations can be eliminated gauge-invariantly, leaving at most second-order equations and fewer than four propagating degrees of freedom.
Weakly Broken Galileon Symmetry
hep-th 2015-04 conditional novelty 6.0 of 10

A broader-than-covariant-galileon subclass of Horndeski theories is identified as the class that maximally preserves galileon non-renormalization once gravity is coupled, and is used to build radiatively stable inflat...
Exploring gravitational theories beyond Horndeski
astro-ph.CO 2014-08 conditional novelty 6.0 of 10

A new class of scalar-tensor theories beyond Horndeski is shown, via a Dirac constraint analysis, to propagate only three degrees of freedom, and to be preserved by gradient-dependent disformal metric transformations.
Black hole hair in generalized scalar-tensor gravity: An explicit example
gr-qc 2014-08 accept novelty 6.0 of 10

A linear coupling between a scalar field and the Gauss-Bonnet invariant evades the no-hair theorem, and explicit numerical black hole solutions show scalar hair plus a minimum black hole size set by the coupling.
Riding on irrelevant operators
hep-th 2014-05 conditional novelty 6.0 of 10

An exact renormalisation-group analysis indicates that P(X) and Galileon scalar theories remain radiatively stable at all loop orders deep in the large-kinetic-term regime, and that the extra DBI symmetry does not ext...
Unification of Galileon Dualities
hep-th 2014-03 accept novelty 6.0 of 10

The most general Galileon duality is a four-parameter GL(2,R) group of coset coordinate transformations, and all known Galileon dualities are special cases of it.
Cosmological perturbations and structure formation in nonlocal infrared modifications of general relativity
astro-ph.CO 2014-03 conditional novelty 6.0 of 10

The R□⁻²R nonlocal dark energy model has stable perturbations and predicts scale-independent deviations from general relativity of a few percent in structure growth and lensing, with growth index γ ≈ 0.53.
Quantum Corrections in Massive Gravity
hep-th 2013-07 conditional novelty 6.0 of 10

At one loop, matter loops leave the ghost-free massive gravity potential intact and graviton loops detune it only in a way that keeps any induced ghost at or above the Planck scale, so the mass structure is technicall...
A naturally light dilaton
hep-th 2013-06 conditional novelty 6.0 of 10

A light dilaton arises naturally, with mass suppressed by a small near-marginal symmetry-breaking parameter, when the underlying theory has a line of fixed points whose quartic coupling crosses zero.
Ghost-Free Derivative Interactions for a Massive Graviton
hep-th 2013-05 conditional novelty 6.0 of 10

Pseudo-linear interaction terms built from anti-symmetrized flat metrics give ghost-free derivative couplings for a massive graviton, and are conjectured to have fully non-linear counterparts beyond dRGT.
A non-local theory of massive gravity
hep-th 2013-05 conditional novelty 6.0 of 10

A covariant non-local massive gravity theory is constructed whose scalar ghost has mass equal to the graviton mass and decouples as m goes to zero, making its vacuum instability negligible for m of order H0.
Effective theory for the Vainshtein mechanism from the Horndeski action
hep-th 2013-05 conditional novelty 6.0 of 10

A scalar-tensor coupling that survives local field redefinitions makes the spherically symmetric vacuum solution inside the Vainshtein radius unstable and superluminal, unless its coefficient is tuned to roughly (rs/r...

See all 116 Pith citations

How later papers used it

254 later papers cite it

Evidence: substantial. 250 of the citing papers share no author with it, and 31 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 445 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

4 later papers by other authors are listed below: 4 used its method or tool.

Uses its method or tool 4 of 8 papers with this label

Computed the mass spectrum including all KK states using the holographic approach of this paper.

Leptons in Holographic Composite Higgs Models with Non-Abelian Discrete Symmetries (2011) · 1.00 · paraphrased: the licence does not allow quoting
Used the holographic approach of this paper, with gauge-fixing outlined in another work, to derive an implicit but relatively compact form for the potential.

Dark Matter and Electroweak Symmetry Breaking in Models with Warped Extra Dimensions (2008) · 1.00 · paraphrased: the licence does not allow quoting
Followed steps completely analogous to those of this paper to calculate the boundary effective action.

Bi-galileon theory I: motivation and formulation (2010) · 0.99 · paraphrased: the licence does not allow quoting
Following this paper, changed to ADM variables in the y direction.

Boundary Terms and Junction Conditions for the DGP Pi-Lagrangian and Galileon (2009) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

237 arXiv papers cite this in 427 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 · 310 sentences without a confident label

Builds on it 2 papers

These shift symmetries generalize the extended shift symmetries of the flat space galileons [ref] to (A)dS space and to higher spins.

Fermionic Shift Symmetries in (Anti) de Sitter Space · Introduction · 0.92
These symmetries generalize the extended shift symmetries of the flat space galileon [ref] and special galileon [ref] to AdS and to higher spins.

Shift Symmetries and AdS/CFT · Introduction · 0.88

Uses its method or tool 8 papers

Following the standard procedure outlined in Refs. [ref] , it is convenient to adopt the ADM-like variables adapted to the DGP case (see [ref] for details), and it can be shown that the action induced for ([ref]) in $4D$ takes the following form [equation] where the gauge-fixing term is based on de Donder gauge, being equal to [equation] and $m≡M_5^3/M_4^2$, where $M_5$ is the mass scale on the bulk and $M_4$ is on the brane is known as the DGP or crossover scale.

Softly higher-derivative massive gravity · Propagator in a softly higher-derivative massive gravity · 0.96
Following [ref] , we change to ADM variables in the $y$ direction (see the appendices of [ref] for conventions on the ADM decomposition).

Boundary Terms and Junction Conditions for the DGP Pi-Lagrangian and Galileon · Derivation of the boundary term from the bulk · 0.98
To calculate the boundary effective action, we essentially follow steps completely analogous to the ones taken in [ref] .

Bi-galileon theory I: motivation and formulation · Example: cascading cosmology · 0.99

Compares with it 8 papers

Indeed, the last term in the first line of (<ref>) is identical to the one found in the decoupling limit of the DGP model [ref] (see, also [ref] for related discussions).

Selftuned Massive Spin-2 · 3. Extracting the longitudinal mode · 0.81
Conceptually, the non-renormalization appears because the tree-level interactions in the Lagrangian are diff invariant up to total derivatives only; on the other hand, the variations of the Lagrangian w.r.t. fields in this theory are exactly diff invariant; therefore, no Feynman diagram can generate operators that would not be diff invariant, and the original operators that are diff invariant only up to total derivatives stay unrenormalized[This is similar to non-renormalization of the Galileon operators [ref] , with diff invariance replaced by galilean invariance; we thank Kurt Hinterbichler for useful discussions on these points.].

Non-Renormalization and Naturalness in a Class of Scalar-Tensor Theories · The Effective Field Theory · 0.83
The $π,ϕ$-Lagrangian associated with Eq. [ref] is the analogue of the $π$-Lagrangian in DGP gravity [ref] .

Bi-galileon theory I: motivation and formulation · Example: cascading cosmology · 0.88

Cites it as support 11 papers

Thanks to the Galileon invariance, the $ℒ_n$'s above are not renormalized under loop corrections, and terms with fewer derivatives acting on the $π$ field are not generated quantum mechanically [ref] (a similar proof holds for the multi-field Galileons [ref] ).

Classical Stability of the Galileon · The Galileon · 0.97
Here we are rescued by galilean invariance, thanks to which the $L_n$'s above do not get renormalized upon loop corrections, and terms with fewer derivatives per field are not generated quantum-mechanically [ref] .

Derrick's theorem beyond a potential · Introduction · 0.98
Secondly, in the Galileon case one can show that there are non-renormalization theorems that prevent the newly defined relevant operators from receiving loop corrections from $π$ loops [ref] .

DBI and the Galileon reunited · Relevant operators and quantum corrections · 0.98

Cites it as background 77 papers

The Galileons [ref] are higher derivative scalar field theories with many interesting and important properties, including second order equations of motion and novel non-linearly realized shift symmetries.

A Covariant Master Theory for Novel Galilean Invariant Models and Massive Gravity · Introduction · 1.00
There exists a range of complicated theories which exhibit the Vainshtein mechanism, but the Galileon [ref] , a model with a single real scalar field, is the simplest example that encapsulates all the essential features, and is known to arise naturally in the decoupling limit of massive gravity theories [ref] .

Simulating a numerical UV Completion of Quartic Galileons · Introduction · 0.95
The simplest model incorporating the Vainshtein mechanism is what is now known as the cubic Galileon model [ref] .

UV properties of Galileons: Spectral Densities · Introduction · 0.91

Discussion

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