Renormalization group flow of projectable Ho\v{r}ava gravity in (3+1) dimensions

Alexander V. Kurov; Andrei O. Barvinsky; Sergey M. Sibiryakov

REVIEW 3 major objections 4 minor 1 cited by

Renormalization group flow of projectable Ho\v{r}ava gravity in (3+1) dimensions

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read Projectable Hořava gravity has exactly one asymptotically free ultraviolet fixed point from which trajectories can reach the general-relativistic regime $\lambda\to 1^+$.

desk verdict A careful one-loop census of the projectable Horava RG landscape that completes the A-to-B-to-λ→1+ picture, but the central hierarchy prediction still rests on one-loop beta functions in a large-coupling regime where control is not demonstrated. read the letter →

arxiv 2411.13574 v1 pith:JM4DZR3D submitted 2024-11-13 gr-qc hep-th

Andrei O. Barvinsky , Alexander V. Kurov , Sergey M. Sibiryakov This is my paper · ORCID

classification gr-qchep-th PACS 04.60.-m11.10.Hi

keywords HoravagravityprojectablerenormalizationgroupflowasymptoticfreedomfixedpointsLifshitzscalingLorentzinvarianceviolationPlanckmasshierarchy

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 asks which renormalization group trajectories a quantum theory of gravity called projectable Hořava gravity can follow, and whether any of them are both consistent in the ultraviolet and look like general relativity at low energies. The authors study the flow of the marginal couplings, the coefficients that stay marginal under the theory's Lifshitz scaling, and claim that among all asymptotically free ultraviolet fixed points only one, located at infinite kinetic coupling $\lambda$, can serve as a source for trajectories that later reach $\lambda\to1^+$, the regime where the kinetic term matches general relativity. Every such trajectory is nearly the same curve in coupling space; only the overall gravitational coupling differs. Along that curve the gravitational coupling first rises, then falls, ending very small in the infrared. Requiring the theory to remain weakly coupled all the way then implies that the Lorentz-violation scale sits far below the Planck mass, a hierarchy that was previously regarded as fine-tuned.

What carries the argument

The machinery is the one-loop $\beta$-function system for the essential couplings of the theory, in which the gravitational coupling $G$ factorizes, so the flow of $\{\lambda, u_s, v_1, v_2, v_3\}$ can be studied independently. The load-bearing objects are two fixed points at infinite $\lambda$: point A, the unique asymptotically free fixed point whose repulsive eigenvectors carry the flow toward finite $\lambda$, and point B, a non-asymptotically-free attractor inside the $\lambda=\infty$ plane. The universal trajectory is the stable manifold from A to B followed by ejection along B's single repulsive eigenvector; the ratio of $\beta$ functions near B yields the steep power law $G \propto \lambda^{3.84}$, which converts the condition of weak coupling into a very small infrared $G$ and hence into the Planck-mass hierarchy. The paper also identifies complex eigenvalues of stability matrices at some fixed points and argues these do not contradict unitarity because the deforming operators are not invariant under the linearized gauge symmetry.

What would settle it

Compute the two-loop $\beta$ functions for the marginal couplings and check whether the fixed points A and B, the connecting stable manifold, and the power-law exponents survive in the large-coupling region near B and in the deep infrared $\lambda\to1^+$. If two-loop corrections change the sign of the $\beta$ function for $G$ near B, move point B, or open a repulsive direction that breaks the trajectory family, the unique-trajectory and hierarchy claims fail. A less expensive check is to repeat the fixed-point search with a different elimination of variables at higher precision, looking specifically for an additional asymptotically free fixed point with $\lambda>1$.

Watch

Extended reading notes

Core claim

The central discovery is that the renormalization group flow of projectable Hořava gravity in (3+1) dimensions has a unique asymptotically free fixed point, called point A, sitting at infinite kinetic coupling $\lambda$, whose repulsive directions reach the physically interesting infrared region $\lambda\to1^+$. Trajectories leaving point A are first attracted to a second fixed point, point B, which is not asymptotically free but governs intermediate scales; near B the flow scatters and then follows the single repulsive direction of B. After the scattering, all viable trajectories coincide in the space of the couplings $\{\lambda, u_s, v_1, v_2, v_3\}$, and their differences reduce to the initial value of the gravitational coupling $G$. The running of $G$ is non-monotonic: it grows after leaving A, reaches a maximum, and then falls, so that its infrared value $G_{\rm IR}$ is extremely small whenever the intermediate maximum is held below one for perturbative consistency. The paper concludes that $M_{\rm LV}/M_{\rm Pl} = \sqrt{G_{\rm IR}} \ll 1$ arises naturally, rather than by accident.

Load-bearing premise

The calculation assumes that quantum corrections computed to one loop stay reliable even where some couplings grow very large, both near the intermediate fixed point called B and in the deep infrared approach to $\lambda=1$, because the claimed unique trajectory and its hierarchy are obtained by integrating the flow through exactly those regions.

Editorial extensions

If this is right

If correct, projectable Hořava gravity has a distinguished UV completion: among all asymptotically free fixed points only point A can reach $\lambda\to1^+$, so phenomenological modeling can focus on a single family of trajectories.
All trajectories that reach the general-relativistic regime pass through the vicinity of point B and then coincide; low-energy physics of the marginal couplings is essentially universal, with the only free data being the normalization of $G$.
The same flow makes $G$ vanish both in the ultraviolet and the infrared, so the theory is asymptotically free in the UV, while in the IR the vanishing of $G$ does not mean the theory is free because the other couplings grow.
Weak coupling along the flow enforces $G_{\rm IR}\ll1$, which translates into $M_{\rm LV}/M_{\rm Pl} = \sqrt{G_{\rm IR}} \ll 1$; the Lorentz-violation scale is naturally many orders of magnitude below the Planck mass.
The numerical search finds no asymptotically free fixed points inside the unitary interval $\lambda>1$ at finite $\lambda$, and the remaining finite-$\lambda$ fixed points all lie at $\lambda<1/3$, so none of them can compete with point A.

Reading between the lines

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

One can infer, beyond the paper, that the same scattering-off-an-intermediate-attractor mechanism may appear in other Lifshitz-type gravity models; if the non-projectable version has an analogous point B, the hierarchy it produces would suppress Lorentz-violating effects in the gravitational sector at low energies.
The one-loop exponent $\kappa=3.84$ is a natural place to test the claim; if a two-loop computation changes the sign of the beta function near B or shifts the stable manifold, the numerical hierarchy would change, so the quantitative prediction is checkable.
The complex stability eigenvalues may be a general feature of gauge theories at asymptotically free fixed points rather than a peculiarity of Hořava gravity; searching for similar complex eigenvalue pairs in other gauge theories would tell whether the unitarity argument generalizes.
Because the hierarchy is tied to the RG time the flow spends near B, an embedding of this model in a larger theory would need no special parameter choice to explain the smallness of Lorentz violation; the flow itself produces it.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Worth a serious look. This paper completes the one-loop fixed-point census for (3+1)-dimensional projectable Horava gravity: five finite-λ points including new F5, eight points at λ=∞, with stability matrices and a thoughtful discussion of why complex eigenvalues need not contradict unitarity. The numerical search is done honestly: multiple equation-removal strategies, asymptotic arguments near λ=1/3 and us→∞, and a Mathematica NSolve cross-check. The IR asymptotics in Appendix C are genuinely useful, with explicit exponents for us, va, and G as λ→1+. The paper also says plainly where it is not venturing: F3–F5 trajectories are excluded because the one-loop approximation is questionable there, and the deep IR is acknowledged as strong-coupled. That transparency earns credit.

Now the main caveat, which the stress-test note puts exactly right. The trajectories that reach λ→1+ do so by passing near fixed point B, where us≈440 and v1≈−1.36×10^4. The one-loop beta functions contain powers up to us^9 and va^3, so the effective expansion parameter is not simply G but G times large-coupling-dependent factors. The condition G<1 does not obviously control those factors, and the paper gives no two-loop estimate or parametric argument that the neglected terms stay small. The claimed hierarchy MLV≪MPl and the exponent κII=3.84 that controls it are obtained by integrating through precisely that region. So the hierarchy result is conditional on an assumption the paper does not discharge. This is not an internal contradiction; the authors are careful and the claim is a legitimate one-loop prediction. But it is a genuine uncontrolled step, and it should be the referee's first question.

Two smaller points. First, the central trajectory family from A to λ→1+ already appeared in the authors' 2023 letter [26]; the new value here is the fixed-point classification, the F3–F5 exclusion, the stability analysis, and the G-running/hierarchy discussion. That is enough to justify the paper, but the novelty is incremental rather than a surprise. Second, fixed-point completeness is a numerical claim—strongly supported, but not an algebraic proof. The multiple scanning strategies and the NSolve check make the list credible, yet a skeptical reader cannot independently verify it without code or data, which are not shipped.

Who gets value from this: people working on Horava gravity and on RG flows of Lifshitz-type theories. It deserves a serious referee. The referee should push for an estimate of two-loop contributions or some scaling argument that controls the large-coupling region near B. If the authors can supply that, this becomes the standard reference for the one-loop global flow. Without it, the hierarchy should be treated as an interesting but unproven one-loop prediction.

Referee Report

3 major / 4 minor

Summary. The paper studies the one-loop renormalization group flow of the marginal couplings of projectable Hořava gravity in (3+1) dimensions. Using the beta-functions of their earlier paper [25], the authors numerically locate all real fixed points of the flow: five at finite kinetic coupling λ and eight at λ = ∞ (Tables I and II). They analyze the stability matrices, showing that some have complex eigenvalues and arguing that this need not contradict unitarity. They then construct RG trajectories emanating from the asymptotically free fixed points. They find that all trajectories from the finite-λ fixed points F1, F2 and from the λ = ∞ fixed points 4 and 7 run into strong coupling, while trajectories from the fixed point A pass near the non-asymptotically-free point B and then flow to the GR-like region λ → 1+ or to the boundary λ → 1/3−, covering the full unitary domain. The gravitational coupling G evolves non-monotonically, and requiring G < 1 along the flow leads to a predicted hierarchy M_LV ≪ M_Pl between the Lorentz-violation scale and the effective Planck mass. The central claims are the uniqueness of the fixed point A for phenomenologically interesting IR behavior and the naturalness of this hierarchy.

Significance. If the one-loop input and the numerical classification are reliable, the paper provides the first complete global picture of the RG flow of projectable Hořava gravity in (3+1) dimensions: a distinguished asymptotically free UV completion (point A), a universal trajectory structure, and a falsifiable prediction that the Lorentz-violation scale is naturally many orders of magnitude below the Planck mass. The strengths of the paper include a thorough fixed-point search with multiple cross-checks (removing different equations, scanning in different variables, direct NSolve verification), explicit stability matrices with a thoughtful discussion of complex eigenvalues, and a transparent derivation of the IR asymptotics in Appendix C. The main risk is that the quantitative hierarchy prediction is obtained by integrating one-loop beta-functions through a region of large couplings near the fixed point B, where the validity of the one-loop approximation is not established and is, in fact, explicitly questioned by the authors for other fixed points with similarly large couplings.

major comments (3)

[§V B, §VI B, Table II, Eq. (38b)] The central hierarchy prediction is obtained by integrating the one-loop beta-functions (16) through the neighborhood of the non-asymptotically-free fixed point B, where us ≈ 440 and v1 ≈ -13566 (Table II). The authors exclude the fixed points F3-F5 from trajectory analysis precisely because such large couplings make the one-loop approximation unreliable (§V A), but no analogous justification is given for the trajectories from A that pass near B. The weak-coupling condition G < 1 invoked in §VI B is insufficient because the beta-functions (16) contain powers up to u_s^9 and v_a^3, so the effective loop-expansion parameter is G multiplied by large polynomials of the couplings, which need not be small even when G < 1. Since the exponent κ_II = 3.84 in Eq. (38b) controls the suppression of G_IR in Eq. (42) and Fig. 11, the predicted hierarchy M_LV/M_Pl = sqrt(G_IR) is not yet quantitatively controlled. Please provide an estimate of two-loop corrections in the large-coupling region, or alternatively state explicitly that the hierarchy is a one-loop-level prediction whose reliability near B and in the deep IR is unproven.
[§III, Appendix A] The paper claims a complete classification of the fixed points, but the evidence is numerical rather than algebraic. The scans are finite (u_t up to 10^8, u_s up to 10^15), and the asymptotic argument in Appendix A only shows that no further solutions exist in the limit at leading order, not as a rigorous proof. The Bezout bound quoted in §III is about 3×10^5 complex roots, so the absence of additional real roots is a numerical inference. Since the uniqueness of the fixed point A for the phenomenologically interesting trajectories is a central claim, the authors should either provide a rigorous certificate (e.g., a Groebner basis computation or interval-arithmetic verification) or rephrase the statements as strong numerical evidence rather than a complete classification.
[§V A, §VII] The statement that 'apart from the family found in [26], all trajectories quickly run into strong coupling' is not supported for the asymptotically free fixed points F3-F5, whose trajectories are explicitly not analyzed in §V A because of large couplings. While those fixed points lie at λ < 1/3 and therefore cannot reach the λ → 1+ region, the 'all trajectories' claim in the Summary is too strong as written. The paper should either analyze the trajectories emanating from F3-F5, or narrow the claim to the set of fixed points for which the one-loop approximation is considered valid.

minor comments (4)

[Throughout] There are several typographical errors: 'differring' in the Introduction, 'reparameteraization' in Section II, 'perurbaation theory' in Section VI B, 'asymptoically' in Section VII, and 'he gravitational coupling' in Section VI A.
[§V B 4] The critical angle δ is used in the bullet list before its numerical value is defined; please move the definition earlier or add a parenthetical remark.
[Abstract and §V B 4] The abstract speaks of a 'single universal trajectory', but the paper describes two distinct universal branches, one to λ → 1+ and one to λ → 1/3− (Figs. 4 and 5). Clarify that 'universal' applies separately to each of the two families.
[§II, Eq. (16)] The explicit polynomials P_G_n and P_χ_n are not reproduced in the paper but only referenced to [25]. For a self-contained reading, at least a brief statement of the maximum degrees and the regime of validity of the one-loop computation would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's central results are numerical consequences of independently stated one-loop beta functions, and the hierarchy claim follows from a weak-coupling consistency condition rather than from a fitted parameter.

full rationale

The paper's derivation chain is self-contained in the relevant sense: the beta functions in Eq. (16) are the input, taken from the same authors' earlier one-loop computation [24,25]. This is a self-citation, but it is not load-bearing in a circular way. The cited beta functions are a parameter-free, explicit set of differential equations whose derivation does not assume any of the present paper's conclusions—neither the classification of fixed points, nor the existence of a universal trajectory, nor the hierarchy MLV << M_Pl. The fixed points in Tables I and II are roots of those beta functions, the stability matrices and exponents (38) are computed from their derivatives, and the scaling relation (42) follows from linearization around the fixed points. No fitted parameter is renamed as a prediction: the value Gmax = 0.25 used in Fig. 11 is a representative choice satisfying the weak-coupling requirement G < 1, and the qualitative conclusion GIR << 1 is robust to this choice because it is controlled by the exponent kappa_II = 3.84, not by the normalization. The deep-IR asymptotics (39)-(40) are derived from the leading Laurent terms of the same beta functions and verified against numerical integration, which is an internal consistency check rather than a circular reduction. The paper's own caveat about one-loop reliability at large couplings near some fixed points (e.g., F3-F5 and point B) is a genuine correctness risk, but it is a quantitative validity concern, not a circularity of the derivation. Therefore no circular step meeting the evidentiary standard can be identified, and the score is 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claims rest on the authors' previously computed one-loop beta functions, on the reliability of those equations in large-coupling regions, and on a finite numerical scan plus asymptotic argument for fixed-point completeness. No new particles, forces, dimensions, or fundamental entities are introduced.

free parameters (3)

initial offset epsilon = 1e-5
Small displacement from fixed points in Eq. (31); chosen by hand to stay in the linearized regime and does not affect qualitative conclusions.
finite scan upper bounds for u_t and u_s = u_t <= 1e8, u_s <= 1e15
Fixed-point completeness is argued by scanning to these finite cutoffs plus asymptotic analysis, not by a rigorous algebraic proof.
maximum gravitational coupling G_max = 0.25 in Fig. 11
Normalization of G trajectories chosen for illustration of G_IR scaling; the hierarchy statement assumes G_max < 1.

assumptions (6)

domain assumption The one-loop beta functions of projectable Horava gravity in 3+1 dimensions computed in [25] are correct.
All fixed points, trajectories, and hierarchy results in this paper are derived from these beta functions; the calculation is cited from the authors' own earlier work and is not independently verified here or in shipped code.
domain assumption Perturbative renormalizability of projectable Horava gravity in any dimension, established in [21,22].
Justifies treating the marginal couplings as a closed set under renormalization; cited as background.
ad hoc to paper One-loop beta functions remain reliable along trajectories with large couplings, especially near fixed point B and in the deep infrared.
The paper excludes F3-F5 because couplings are large, but still uses one-loop equations for the A-to-B segment (v1 near -13600, us near 440) and near lambda to 1; no quantitative criterion for one-loop validity is given.
ad hoc to paper The finite numerical scan plus the asymptotic argument exhausts all fixed points.
Completeness is asserted on the basis of residual-crossing scans in u_t and u_s intervals with specified steps, plus a large-us asymptotic argument; no algebraic certificate of completeness is provided.
domain assumption The unitary domain lambda < 1/3 or lambda > 1, together with positivity conditions on nu5 and related couplings.
Restricts the allowed parameter space and defines which trajectories are physically relevant; introduced in Section II Eq. (10).
standard math Higher-loop contributions are suppressed at asymptotically free fixed points because the beta functions vanish at G = 0.
Standard consequence of the G factorization of the beta functions; used to argue that the fixed-point locations are robust beyond one loop.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Renormalization group flow of projectable Ho\v{r}ava gravity in (3+1) dimensions." pith.science (2026). https://pith.science/paper/JM4DZR3D

@misc{pith2026241113574,
  author       = {Pith},
  title        = {Pith review of: Renormalization group flow of projectable Ho\vrava gravity in (3+1) dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JM4DZR3D}},
  note         = {Machine review of arXiv:2411.13574}
}

abstract

We report a comprehensive numerical study of the renormalization group flow of marginal couplings in $(3+1)$-dimensional projectable Ho\v{r}ava gravity. First, we classify all fixed points of the flow and analyze their stability matrices. We find that some of the stability matrices possess complex eigenvalues and discuss why this does not contradict unitarity. Next, we scan over the renormalization group trajectories emanating from all asymptotically free fixed points. We identify a unique fixed point giving rise to a set of trajectories spanning the whole range of the kinetic coupling $\lambda$ compatible with unitarity. This includes the region $0<\lambda-1\ll 1$ assumed in previous phenomenological applications. The respective trajectories closely follow a single universal trajectory, differing only by the running of the gravitational coupling. The latter exhibits non-monotonic behavior along the flow, vanishing both in the ultraviolet and the infrared limits. The requirement that the theory remains weakly coupled along the renormalization group trajectory implies a natural hierarchy between the scale of Lorentz invariance violation and a much larger value of the Planck mass inferred from low-energy interactions.

Figures

Figures reproduced from arXiv: 2411.13574 by the authors.

**Figure 2.** FIG. 2. RG flow from fixed point F2 (red dot) initiated along [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 4.** FIG. 4. Couplings [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

Figures from the paper (11 more)

**Figure 6.** Figure 6: FIG. 6. A chart illustrating global properties of the RG trajec [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: FIG. 7. RG trajectory from fixed point A to [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: FIG. 8. RG trajectory from fixed point A to [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: FIG. 9. Behavior of [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: FIG. 10. Behavior of [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]

**Figure 11.** Figure 11: FIG. 11. Dependence of the IR value of the gravitational cou [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]

**Figure 12.** Figure 12: FIG. 12. Dependence of the residual [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]

**Figure 13.** Figure 13: FIG. 13. Behavior of the residual [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]

**Figure 14.** Figure 14: FIG. 14. RG flow from fixed point 4 (red dot). Dashed black [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]

**Figure 16.** Figure 16: FIG. 16. Beta-function of the coupling [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]

**Figure 17.** Figure 17: FIG. 17. Same as Fig. 16 but for the beta-functions of the couplings [PITH_FULL_IMAGE:figures/full_fig_p021_17.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

HBT Interferometry and Quantum Nature of Primordial Gravitational Waves in Ho\v{r}ava-Lifshitz Gravity
gr-qc 2024-12 conditional novelty 6.0 of 10

In Hořava-Lifshitz gravity, primordial gravitational waves can be generated without inflation and their nonclassical statistics might be detectable by HBT interferometry above about 10 kHz in the radiation era or 10^-...

Reference graph

Works this paper leans on

69 extracted references · 29 canonical work pages · cited by 1 Pith paper

[26]

Pujolas and S

O. Pujolas and S. Sibiryakov, Supersymmetric Aether, JHEP 01, 062, arXiv:1109.4495 [hep-th]

arXiv
[25]

Pospelov and Y

M. Pospelov and Y. Shang, On Lorentz violation in Horava-Lifshitz type theories, Phys. Rev. D 85, 105001 22 (2012), arXiv:1010.5249 [hep-th]

arXiv 2012
[1]

The components of the cor- responding eigenvectors w1, w2 are presented in the first two rows of Table V (labeled FP1 w1 and FP1w2)

Flow from fixed point F1 According to Table III the first fixed point has two neg- ative eigenvalues θ1 and θ2. The components of the cor- responding eigenvectors w1, w2 are presented in the first two rows of Table V (labeled FP1 w1 and FP1w2). Note that these eigenvectors are almost collinear and aligned along the with v1-direction. In the initial condit...

work page
[2]

The components of the corre- sponding eigenvector w1 are given in the third row of Table V (labeled FP2w1)

Flow from fixed point F2 According to Table III, the second fixed point has only one repulsive direction. The components of the corre- sponding eigenvector w1 are given in the third row of Table V (labeled FP2w1). Note that, as in the case of the first fixed point, this eigenvector nearly aligns along one of the coordinate axes — us-direction in this case...

work page 2000
[3]

Flow between fixed points A and B The fixed point A has two negative eigenvalues, see Table IV. The components of the corresponding repulsive eigenvectors are listed in the first two rows of Table VI (labeled wA1 and wA2).9 The initial conditions for the RG trajectories are set according to Eq. (31) with cA1wA1 + cA2wA2 = cos φAwA1 + sin φAwA2 , (33) wher...

work page
[4]

Thus, it cannot serve as a UV fixed point of the RG flow

Flow from point B to λ → 1+ The fixed point B is not asymptotically free. Thus, it cannot serve as a UV fixed point of the RG flow. How- ever, as we saw above, it can play the role of an attractor at intermediate RG scales. The flow leaves the attractor along the unique repulsive eigenvectorwB1 whose compo- nents are listed in the third row of Table VI. T...

work page
[5]

The couplings us, v1 grow rapidly in absolute value along the trajectory, while v2, v3 are of order 1 until the trajectory reaches vicinity of λ = 1/3

Flow from point B to λ → 1/3− On the trajectory with cB1 = 1 the coupling ϱ mono- tonically increases and at τ → −∞ reaches another boundary of the unitarity domain (10a) ϱ → ∞(λ → 1/3−). The couplings us, v1 grow rapidly in absolute value along the trajectory, while v2, v3 are of order 1 until the trajectory reaches vicinity of λ = 1/3. Despite large val...

work page
[6]

For different values of φA we obtain the follow- ing global behavior of the RG trajectories schematically illustrated in Fig

Flow from fixed point A to λ = O(1) We now consider a general linear combination of vec- tors wA1 and wA2 in the initial condition (33) at the point A. For different values of φA we obtain the follow- ing global behavior of the RG trajectories schematically illustrated in Fig. 6: • φA ∈ [δ, π 2 ), where δ ≪ 1:11 the trajectory passes in the neighborhood o...

work page

Show all 69 references

[7]

III A, we eliminate λ from the system of equations for the fixed points using Eq

Finite λ As described in Sec. III A, we eliminate λ from the system of equations for the fixed points using Eq. (20) and introduce the variable ut according to Eq. (22). We scan over the values u∗ t in the interval from √ 10 to 10 8 with different steps ϵ in different interval...
[8]

Infinite λ Beta-functions for the variables χ = {us, va} in the limit λ → ∞(ϱ = 1) have the form, ˜βχ ϱ=1 = ¯Aχ 26880π2(1 + us)3u5s 9X n=0 un s S χ n [va] , ¯Aus = us , ¯Av1 = 1 , ¯Av2 = ¯Av3 = 2 , (A1) where the polynomials S χ n [va] are obtained from the poly- nomials P χ n...
[9]

The components of the corresponding eigen- vectors wJ are collected in three upper rows of Table VII (labeled w4|1, w4|2, w4|3)

Flow from point №4 Stability matrix at the point 4 has three negative eigen- values θJ . The components of the corresponding eigen- vectors wJ are collected in three upper rows of Table VII (labeled w4|1, w4|2, w4|3). Note that the eigenvectors are almost collinear and nearly ...
[10]

ϑ ∈ [0, π/2), φ-any: the projections of the trajec- tory on different planes are shown by black dashed curves in Fig. 14
[11]

ϑ ∈ [π/2, π], φ-any: the trajectory is shown by the blue solid curves in Fig. 14. The trajectories cannot be continued any further be- cause of the loss of numerical precision due to a rapid growth of v1. In other words, the trajectories run into singularity in a finite ‘RG ti...
[12]

Two of them (θ1, θ2) are real and two ( θ3, θ4) are complex con- jugate

Flow from point №7 Fixed point 7 has four negative eigenvalues θJ . Two of them (θ1, θ2) are real and two ( θ3, θ4) are complex con- jugate. The repulsive vectors are listed in the four lower rows of Table VII (labeled w7|1 through w7|4). The vec- tors w7|1 and w7|2 are eigenv...
[13]

ϑ ∈ [0, π/2), ψ-any, φ-any: the projections of the trajectory on different planes are shown by black dashed curves in Fig. 15
[14]

ϑ ∈ [π/2, π], ψ-any, φ-any: the trajectory is shown by the blue solid curves in Fig. 15. The trajectories cannot be continued any further be- cause of the loss of precision due to a rapid growth of the couplings in absolute values. We conclude that the tra- jectories run into ...

2000
[15]

Hoˇ rava, Quantum Gravity at a Lifshitz Point, Phys

P. Hoˇ rava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D79, 084008 (2009), arXiv:0901.3775 [hep-th]

2009 arXiv
[16]

Mukohyama, Hoˇ rava-Lifshitz Cosmology: A Review, Class

S. Mukohyama, Hoˇ rava-Lifshitz Cosmology: A Review, Class. Quant. Grav. 27, 223101 (2010), arXiv:1007.5199 [hep-th]

2010 arXiv
[17]

T. P. Sotiriou, Hoˇ rava-Lifshitz gravity: a status report, J. Phys. Conf. Ser. 283, 012034 (2011), arXiv:1010.3218 [hep-th]

2011 arXiv
[18]

A. O. Barvinsky, Hoˇ rava Models as Palladium of Uni- tarity and Renormalizability in Quantum Gravity (2023) arXiv:2301.13580 [hep-th]

2023 arXiv
[19]

Herrero-Valea, The status of Hoˇ rava gravity, Eur

M. Herrero-Valea, The status of Hoˇ rava gravity, Eur. Phys. J. Plus 138, 968 (2023), arXiv:2307.13039 [gr-qc]

2023 arXiv
[20]

K. S. Stelle, Renormalization of Higher Derivative Quan- tum Gravity, Phys. Rev. D16, 953 (1977)

1977
[21]

K. S. Stelle, Classical Gravity with Higher Derivatives, Gen. Rel. Grav. 9, 353 (1978)

1978
[22]

V. A. Kostelecky and N. Russell, Data Tables for Lorentz and CPT Violation, Rev. Mod. Phys. 83, 11 (2011), arXiv:0801.0287 [hep-ph]

2011
[23]

Liberati, Tests of Lorentz invariance: a 2013 update, Class

S. Liberati, Tests of Lorentz invariance: a 2013 update, Class. Quant. Grav. 30, 133001 (2013), arXiv:1304.5795 [gr-qc]

2013 arXiv
[24]

Groot Nibbelink and M

S. Groot Nibbelink and M. Pospelov, Lorentz violation in supersymmetric field theories, Phys. Rev. Lett. 94, 081601 (2005), arXiv:hep-ph/0404271 [hep-ph]

2005 arXiv
[27]

M. M. Anber and J. F. Donoghue, The Emergence of a universal limiting speed, Phys. Rev. D83, 105027 (2011), arXiv:1102.0789 [hep-th]

2011 arXiv
[28]

Bednik, O

G. Bednik, O. Pujolas, and S. Sibiryakov, Emergent Lorentz invariance from Strong Dynamics: Holographic examples, JHEP 11, 064, arXiv:1305.0011 [hep-th]

arXiv
[29]

Kharuk and S

I. Kharuk and S. M. Sibiryakov, Emergent Lorentz in- variance with chiral fermions, Theor. Math. Phys. 189, 1755 (2016), [Teor. Mat. Fiz. 189, no.3, 405 (2016)], arXiv:1505.04130 [hep-th]

2016 arXiv
[30]

Baggioli, O

M. Baggioli, O. Pujolas, and X.-M. Wu, Holographic Lif- shitz flows, JHEP 09, 175, arXiv:2407.11552 [hep-th]

arXiv
[31]

D. Blas, O. Pujolas, and S. Sibiryakov, Consistent Ex- tension of Hoˇ rava Gravity, Phys. Rev. Lett.104, 181302 (2010), arXiv:0909.3525 [hep-th]

2010 arXiv
[32]

Blas and E

D. Blas and E. Lim, Phenomenology of theories of gravity without Lorentz invariance: the preferred frame case, Int. J. Mod. Phys. D23, 1443009 (2015), arXiv:1412.4828 [gr- qc]

2015 arXiv
[33]

Emir G¨ umr¨ uk¸ c¨ uo˘ glu, M

A. Emir G¨ umr¨ uk¸ c¨ uo˘ glu, M. Saravani, and T. P. Sotiriou, Hoˇ rava gravity after GW170817, Phys. Rev. D 97, 024032 (2018), arXiv:1711.08845 [gr-qc]

2018 arXiv
[34]

Bellor ´ ın, C

J. Bellor ´ ın, C. B´ orquez, and B. Droguett, Cancellation of divergences in the nonprojectable hoˇ rava theory, Phys. Rev. D 106, 044055 (2022), arXiv:2207.08938 [hep-th]

2022 arXiv
[35]

A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. Sibiryakov, and C. F. Steinwachs, Renormalization of Hoˇ rava gravity, Phys. Rev. D93, 064022 (2016), arXiv:1512.02250 [hep-th]

2016 arXiv
[36]

A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. Sibiryakov, and C. F. Steinwachs, Renormalization of gauge theories in the background-field approach, JHEP 07, 035, arXiv:1705.03480 [hep-th]

arXiv
[37]

A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. Sibiryakov, and C. F. Steinwachs, Hoˇ rava Gravity is Asymptotically Free in 2 + 1 Dimensions, Phys. Rev. Lett. 119, 211301 (2017), arXiv:1706.06809 [hep-th]

2017 arXiv
[38]

A. O. Barvinsky, M. Herrero-Valea, and S. M. Sibiryakov, Towards the renormalization group flow of Hoˇ rava grav- ity in (3 + 1) dimensions, Phys. Rev. D 100, 026012 (2019), arXiv:1905.03798 [hep-th]

2019 arXiv
[39]

A. O. Barvinsky, A. V. Kurov, and S. M. Sibiryakov, Beta functions of (3+1)-dimensional projectable Hoˇ rava grav- ity, Phys. Rev. D 105, 044009 (2022), arXiv:2110.14688 [hep-th]

2022 arXiv
[40]

A. O. Barvinsky, A. V. Kurov, and S. M. Sibiryakov, Asymptotic freedom in (3+1)-dimensional projectable Hoˇ rava gravity: Connecting the ultraviolet and in- frared domains, Phys. Rev. D 108, L121503 (2023), arXiv:2310.07841 [hep-th]

2023 arXiv
[41]

Mukohyama, Dark matter as integration constant in Horava-Lifshitz gravity, Phys

S. Mukohyama, Dark matter as integration constant in Horava-Lifshitz gravity, Phys. Rev. D 80, 064005 (2009), arXiv:0905.3563 [hep-th]

2009 arXiv
[42]

Izumi and S

K. Izumi and S. Mukohyama, Stellar center is dynami- cal in Horava-Lifshitz gravity, Phys. Rev. D 81, 044008 (2010), arXiv:0911.1814 [hep-th]

2010 arXiv
[43]

Izumi and S

K. Izumi and S. Mukohyama, Nonlinear superhorizon perturbations in Hoˇ rava-Lifshitz gravity, Phys. Rev. D84, 064025 (2011), arXiv:1105.0246 [hep-th]

2011 arXiv
[44]

A. E. G¨ umr¨ uk¸ c¨ uo˘ glu, S. Mukohyama, and A. Wang, General relativity limit of Hoˇ rava-Lifshitz gravity with a scalar field in gradient expansion, Phys. Rev. D85, 064042 (2012), arXiv:1109.2609 [hep-th]

2012 arXiv
[45]

P. M. Bassani, J. Magueijo, and S. Mukohyama, Vio- lations of energy conservation in Horava-Lifshitz grav- ity: a new ingredient in the dark matter puzzle (2024), arXiv:2408.03793 [gr-qc]

2024 arXiv
[46]

Koyama and F

K. Koyama and F. Arroja, Pathological behaviour of the scalar graviton in Hoˇ rava-Lifshitz gravity, JHEP03, 061, arXiv:0910.1998 [hep-th]

1998 arXiv
[47]

D. Blas, O. Pujolas, and S. Sibiryakov, Models of non- relativistic quantum gravity: The Good, the bad and the healthy, JHEP 04, 018, arXiv:1007.3503 [hep-th]

arXiv
[48]

Germani, A

C. Germani, A. Kehagias, and K. Sfetsos, Relativistic Quantum Gravity at a Lifshitz Point, JHEP 09, 060, arXiv:0906.1201 [hep-th]

arXiv
[49]

D. Blas, O. Pujolas, and S. Sibiryakov, On the Extra Mode and Inconsistency of Hoˇ rava Gravity, JHEP 10, 029, arXiv:0906.3046 [hep-th]

arXiv
[50]

Kimpton and A

I. Kimpton and A. Padilla, Lessons from the decoupling limit of Horava gravity, JHEP 07, 014, arXiv:1003.5666 [hep-th]

arXiv
[51]

B. P. Abbott et al. (LIGO Scientific, Virgo, Fermi- GBM, INTEGRAL), Gravitational Waves and Gamma- rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A, Astrophys. J. 848, L13 (2017), arXiv:1710.05834 [astro-ph.HE]

2017 arXiv
[52]

Kimpton and A

I. Kimpton and A. Padilla, Matter in Hoˇ rava-Lifshitz gravity, JHEP 04, 133, arXiv:1301.6950 [hep-th]

arXiv
[53]

T. P. Sotiriou, M. Visser, and S. Weinfurtner, Phe- nomenologically viable Lorentz-violating quantum grav- ity, Phys. Rev. Lett.102, 251601 (2009), arXiv:0904.4464 [hep-th]

2009 arXiv
[54]

B. S. DeWitt, Quantum theory of gravity. ii. the mani- festly covariant theory, Phys. Rev. 162, 1195 (1967)

1967
[55]

Kallosh, Renormalization in non-abelian gauge theo- ries, Nuclear Physics B 78, 293 (1974)

R. Kallosh, Renormalization in non-abelian gauge theo- ries, Nuclear Physics B 78, 293 (1974)

1974
[56]

A. E. G¨ umr¨ uk¸ c¨ uo˘ glu and S. Mukohyama, Hoˇ rava-Lifshitz gravity with λ → ∞, Phys. Rev. D 83, 124033 (2011), arXiv:1104.2087 [hep-th]

2011 arXiv
[57]

J. I. Radkovski and S. M. Sibiryakov, Scattering ampli- tudes in high-energy limit of projectable Hoˇ rava gravity, Phys. Rev. D 108, 046017 (2023), arXiv:2306.00102 [hep- th]

2023 arXiv
[58]

Buchberger and M

B. Buchberger and M. Kauers, Groebner basis, Scholar- pedia 5, 7763 (2010)

2010
[59]

I. R. Shafarevich, Basic Algebraic Geometry (Springer- Verlag Berlin, Heidelberg, 1977)

1977
[60]

Wolfram Research, Inc., Mathematica, Version 12.2, champaign, IL, 2020

2020
[61]

Komargodski and A

Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP 2011 (12), 99, arXiv:1107.3987 [hep-th]

2011 arXiv
[62]

M. A. Luty, J. Polchinski, and R. Rattazzi, The a- theorem and the asymptotics of 4d quantum field theory, JHEP 2013 (1), 152, arXiv:1204.5221 [hep-th]

2013 arXiv
[63]

Mack, All unitary ray representations of the confor- mal group SU(2,2) with positive energy, Commun

G. Mack, All unitary ray representations of the confor- mal group SU(2,2) with positive energy, Commun. Math. Phys. 55, 1 (1977)

1977
[64]

Nishida and D

Y. Nishida and D. T. Son, Unitary Fermi gas, ep- silon expansion, and nonrelativistic conformal field theo- ries, Lect. Notes Phys. 836, 233 (2012), arXiv:1004.3597 [cond-mat.quant-gas]

2012 arXiv
[65]

W. D. Goldberger, Z. U. Khandker, and S. Prabhu, OPE convergence in non-relativistic conformal field theories, 23 JHEP 12, 048, arXiv:1412.8507 [hep-th]

arXiv
[66]

Hogervorst, S

M. Hogervorst, S. Rychkov, and B. C. van Rees, Unitarity violation at the Wilson-Fisher fixed point in 4- ϵ dimen- sions, Phys. Rev. D 93, 125025 (2016), arXiv:1512.00013 [hep-th]

2016 arXiv
[67]

Gorbenko, S

V. Gorbenko, S. Rychkov, and B. Zan, Walking, Weak first-order transitions, and Complex CFTs, JHEP 10, 108, arXiv:1807.11512 [hep-th]

arXiv
[68]

Gromov, V

N. Gromov, V. Kazakov, G. Korchemsky, S. Negro, and G. Sizov, Integrability of Conformal Fishnet Theory, JHEP 01, 095, arXiv:1706.04167 [hep-th]

arXiv
[69]

C. B. Jepsen, I. R. Klebanov, and F. K. Popov, RG limit cycles and unconventional fixed points in per- turbative QFT, Phys. Rev. D 103, 046015 (2021), arXiv:2010.15133 [hep-th]

2021 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [26]

Pujolas and S

O. Pujolas and S. Sibiryakov, Supersymmetric Aether, JHEP 01, 062, arXiv:1109.4495 [hep-th]

arXiv

[2] [25]

Pospelov and Y

M. Pospelov and Y. Shang, On Lorentz violation in Horava-Lifshitz type theories, Phys. Rev. D 85, 105001 22 (2012), arXiv:1010.5249 [hep-th]

arXiv 2012

[3] [1]

The components of the cor- responding eigenvectors w1, w2 are presented in the first two rows of Table V (labeled FP1 w1 and FP1w2)

Flow from fixed point F1 According to Table III the first fixed point has two neg- ative eigenvalues θ1 and θ2. The components of the cor- responding eigenvectors w1, w2 are presented in the first two rows of Table V (labeled FP1 w1 and FP1w2). Note that these eigenvectors are almost collinear and aligned along the with v1-direction. In the initial condit...

work page

[4] [2]

The components of the corre- sponding eigenvector w1 are given in the third row of Table V (labeled FP2w1)

Flow from fixed point F2 According to Table III, the second fixed point has only one repulsive direction. The components of the corre- sponding eigenvector w1 are given in the third row of Table V (labeled FP2w1). Note that, as in the case of the first fixed point, this eigenvector nearly aligns along one of the coordinate axes — us-direction in this case...

work page 2000

[5] [3]

Flow between fixed points A and B The fixed point A has two negative eigenvalues, see Table IV. The components of the corresponding repulsive eigenvectors are listed in the first two rows of Table VI (labeled wA1 and wA2).9 The initial conditions for the RG trajectories are set according to Eq. (31) with cA1wA1 + cA2wA2 = cos φAwA1 + sin φAwA2 , (33) wher...

work page

[6] [4]

Thus, it cannot serve as a UV fixed point of the RG flow

Flow from point B to λ → 1+ The fixed point B is not asymptotically free. Thus, it cannot serve as a UV fixed point of the RG flow. How- ever, as we saw above, it can play the role of an attractor at intermediate RG scales. The flow leaves the attractor along the unique repulsive eigenvectorwB1 whose compo- nents are listed in the third row of Table VI. T...

work page

[7] [5]

The couplings us, v1 grow rapidly in absolute value along the trajectory, while v2, v3 are of order 1 until the trajectory reaches vicinity of λ = 1/3

Flow from point B to λ → 1/3− On the trajectory with cB1 = 1 the coupling ϱ mono- tonically increases and at τ → −∞ reaches another boundary of the unitarity domain (10a) ϱ → ∞(λ → 1/3−). The couplings us, v1 grow rapidly in absolute value along the trajectory, while v2, v3 are of order 1 until the trajectory reaches vicinity of λ = 1/3. Despite large val...

work page

[8] [6]

For different values of φA we obtain the follow- ing global behavior of the RG trajectories schematically illustrated in Fig

Flow from fixed point A to λ = O(1) We now consider a general linear combination of vec- tors wA1 and wA2 in the initial condition (33) at the point A. For different values of φA we obtain the follow- ing global behavior of the RG trajectories schematically illustrated in Fig. 6: • φA ∈ [δ, π 2 ), where δ ≪ 1:11 the trajectory passes in the neighborhood o...

work page