Pith. sign in

REVIEW 4 major objections 4 minor 40 references

Renormalization-Group Running Induced Cosmic Inflation

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read Renormalization-group running of the inflaton potential can release a false vacuum and trigger slow-roll inflation, removing the need for large GUT-scale field fluctuations.

desk verdict A serious scenario proposal that uses RG convexification to release a false vacuum and start inflation, with honest caveats and a heuristic time–scale identification that needs work. read the letter →

arxiv 1909.00580 v2 pith:BQWUPJ4G submitted 2019-09-02 astro-ph.CO hep-th

I. G. Marian , N. Defenu , U. D. Jentschura , A. Trombettoni , I. Nandori This is my paper · ORCID
classification astro-ph.COhep-th PACS 98.80.Cq
keywords cosmicinflationinitialconditionproblemrenormalizationgroupmassivesine-Gordonmodelslow-rollfalsevacuumconvexityPlanckdata
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 proposes that the inflaton field does not need a specially prepared, large-amplitude initial condition: it can be trapped in a false vacuum at Planck-scale energies and then released by renormalization-group (RG) running. Using the massive sine-Gordon (MSG) potential as a concrete renormalizable example, the authors show that quantum fluctuations flatten the potential as the RG scale runs from the Planck scale down to the scale of inflation, so the false vacuum disappears and the field rolls to the true minimum. Parameters fixed at the inflation scale by a slow-roll analysis fall in regions consistent with Planck data, with the Fourier amplitude $u_k$ decreasing by a factor of about four or more in the favored large-frequency regime. If correct, the mechanism gives a particle-physics origin for large-field, chaotic inflation and removes the need for GUT-scale field fluctuations. The proposed mechanism is argued to apply to any differentiable inflationary potential with a concave region and at least one false vacuum.

What carries the argument

The central object is the RG-flow equation for the effective potential in the local potential approximation with the Litim regulator, applied to the massive sine-Gordon (MSG) model $V(\phi)=\tfrac12 m^2\phi^2+u[1-\cos(\beta\phi)]$. The key result is that only the Fourier amplitude $u_k$ runs; in $d=4$ dimensions the linearized flow gives $u_k=u_\Lambda\exp[\beta^2(k^2-\Lambda^2)/(64\pi^2)]$, so $u_k$ shrinks as the scale drops. The paper identifies the running scale $k$ with the inverse cosmological time ($k\sim1/t\sim H$), so the RG evolution from the Planck scale to the inflation scale describes the actual pre-inflationary history. The convexification of the effective potential under RG flow, proved in Appendix C via the Legendre-transform identity $(\delta^2 V_{\rm eff}/\delta\phi^2)(\delta^2 w/\delta J^2)=1$, is what erases the false vacuum and triggers the roll.

What would settle it

Compute the MSG potential's RG flow in an explicit FLRW background using the same functional RG scheme but with the cosmic time built in through the metric, rather than assumed via $k\sim1/t$; if the false vacuum still traps the field at the would-be inflation scale, the mechanism fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that pre-inflationary quantum fluctuations, treated through the functional renormalization group, can supply the missing initial conditions for slow-roll inflation. For the massive sine-Gordon potential $V(\phi)=\tfrac12 m^2\phi^2+u[1-\cos(\beta\phi)]$, the dimensionful mass $m$ and frequency $\beta$ do not run, but the Fourier amplitude $u_k$ does: between the Planck scale $\Lambda_p$ and the inflation scale $k_i$, $u_k$ decreases, and for $\hat\beta\gtrsim30$ the ratio $u_{\Lambda_p}/u_{k_i}\gtrsim4.2$. This running reshapes the potential from a concave form with a trapping false vacuum at high energies to a flatter, closer-to-convex form at lower energies, releasing the vacuum expectation value to initiate slow roll. The parameters at $k_i$ are fixed by matching the slow-roll predictions for the scalar tilt $n_s$ and tensor-to-scalar ratio $r$ to Planck data; the scale of inflation can be commensurate with the GUT scale (small $\hat\beta$) or lower, around $2\times10^{13}$ GeV (large $\hat\beta$). The authors argue the mechanism is generic: any differentiable inflationary potential with a concave region and a false vacuum should undergo the same RG-driven convexification.

Load-bearing premise

The argument depends on the assumption that the renormalization-group scale moves with cosmic time (roughly as the inverse Hubble time), so the calculated flattening of the potential happens before inflation begins; the paper adopts this identification heuristically rather than proving it.

Editorial extensions

If this is right

  • The starting point of slow roll is fixed by the UV shape of the potential and the RG flow, so different spatial regions begin inflation with the same vacuum expectation value rather than from random large fluctuations.
  • Large-field ("chaotic") inflation can be supported by particle physics: the field is trapped at high energies and released by running, so no super-Planckian field excursion or GUT-scale fluctuation is required.
  • For the MSG model, matching to Planck data yields two viable regimes: small $\hat\beta$ with inflation near the GUT scale, and large $\hat\beta$ (gtrsim30) with inflation near $2\times10^{13}$ GeV and a factor of at least about 4.2 drop in $u_k$ from the Planck scale.
  • The mechanism is argued to be generic: any differentiable inflationary potential with a concave region and at least one false vacuum will convexify under RG flow and release its vacuum expectation value, and adding a constant field-independent term leaves the slow-roll results unchanged.

Reading between the lines

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

  • If the $k\sim1/t$ identification is later derived rather than assumed, the same running would tie the inflaton's initial condition to the dynamics of the field-independent term $V_k(0)$, which the paper identifies with the cosmological constant; a single RG flow could then connect inflation to dark energy.
  • The convexification mechanism could be probed in laboratory analogues, for example ultracold atoms in optical lattices realizing sine-Gordon-type potentials, where coarse graining can be controlled and the disappearance of metastable minima observed.
  • The requirement $\hat\beta\gtrsim30$ for a substantial change in $u_k$ implies a testable hierarchy: if future B-mode measurements pin down $r$ and the inflation scale, they will either select the large-$\beta$ branch or force a super-Planckian running that the paper itself regards as problematic.
  • Because the MSG potential's Taylor expansion reproduces the bi-quadratic Higgs-like form, the same RG-released false-vacuum mechanism could, in principle, supply initial conditions for Higgs-inflation scenarios without invoking a separate inflaton field.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper proposes a mechanism in which the RG running of an inflaton potential between the Planck scale and the GUT scale releases a field trapped in a false vacuum and thereby supplies the initial condition for slow-roll inflation. The authors analyze the massive sine-Gordon (MSG) model of Eq. (2.4) with the local-potential approximation and Litim regulator, derive flow equations for the parameters (Eqs. (3.4)-(3.7)), and obtain an approximate exponential solution for the Fourier amplitude u(k), Eq. (3.10). The parameters are matched to Planck data through a slow-roll analysis at the inflation scale in Secs. 3.3-3.5, and it is argued that for dimensionless frequency beta-hat >= 30 the Fourier amplitude changes by a factor chi >= 4.2 between the Planck and inflation scales (Eq. (3.24)), so that the false vacuum disappears and the field rolls down. The paper includes appendices on the k-time relation, alternative MSG variants, the phi^6 model, convexity of the effective potential, and the field-independent constant term.

Significance. If the proposed mechanism were established, it would give a particle-physics origin for the initial conditions of large-field inflation and would connect the Planck-scale shape of the scalar potential to observed CMB parameters. The paper has real strengths: it uses an explicit nonperturbative RG setup, provides an analytic solution for u(k), and performs careful slow-roll fits against Planck 2015 data in several potential variants. The appendices are informative, and the authors are transparent about many limitations, including the heuristic status of the k-time identification and the flat-space truncation. Nevertheless, the two central premises--the temporal interpretation of the RG scale and the flat-space single-mode treatment of the flow--are exactly the points on which the mechanism depends, so the present version is best read as a scenario for inflation initial conditions rather than a demonstrated derivation.

major comments (4)
  1. [Sec. 2.2; Appendix A] The mechanism's temporal axis is the identification of the RG momentum k with the inverse cosmological time or Hubble scale, which the paper itself describes as 'heuristically assumed' in Sec. 2.2 and 'permissible to associate' in Appendix A. Equations (3.1)-(3.10) are flat-space Euclidean flow equations and contain no time variable; a scale-dependent potential is not automatically a time-dependent potential. Since the claim that the VeV remains trapped while the universe expands from the Planck to the GUT scale depends entirely on this identification, the paper must either derive k(t) from the cosmological dynamics or explicitly reformulate the mechanism as conditional on an unproven identification. This is a load-bearing gap, not a presentation issue.
  2. [Secs. 2.4, 3.1, 3.5, and Eq. (3.24)] The quantitative estimate that the Fourier amplitude changes by chi >= 4.2 uses the flat-space LPA flow in the single-Fourier-mode truncation, Eq. (3.6), together with the linearized solution Eq. (3.10). The effective potential felt by a homogeneous inflaton in an FLRW background need not equal this flat-space blocking result: the curved-space mode functions, the initial state, and the regulator all differ, and the time-dependent rescaling in Eqs. (2.8)-(2.9) generates additional terms involving the time derivative of the scale factor that are not included in the flat-space action of Eq. (2.5). The appeal to Fig. 9 of Ref. [14] in Sec. 2.2 is only qualitative, and Sec. 4 concedes that no realistic GUT calculation is performed. The factor chi should therefore be presented as a toy-model estimate unless a curved-space RG computation is supplied.
  3. [Secs. 3.1, 3.4, and 3.5] The parameters (m0, u0, beta0) are fixed by the slow-roll fit to Planck data at the inflation scale, Eq. (3.11), and the UV value u_Lambda is then obtained by inverse RG evolution via Eq. (3.10). The existence of a false vacuum at the Planck scale is therefore a consistency condition of the fitted IR potential, not an independent prediction from particle physics. The statement in Sec. 4 that the method 'determines a unique initial value' is correspondingly weakened: the initial value is determined by the fit, and the freedom in beta-hat is later used in Sec. 3.5 to make the UV change substantial (beta-hat >= 30). The paper should distinguish more sharply between a consistency check and a prediction, and should identify which observables, if any, would falsify the proposed mechanism.
  4. [Appendix C and Sec. 3.2] The convexity argument in Appendix C shows that the exact effective potential is convex in the k -> 0 limit, but it does not prove that the false vacuum disappears at the finite scale k_i = 2 x 10^13 GeV at which inflation is claimed to begin. The release of the VeV at that scale relies on the quantitative running of u(k), which in turn depends on the linearization of Eq. (3.7) and on the single-mode truncation of Eq. (3.6). The paper should state this finite-scale limitation explicitly and, ideally, demonstrate with the untruncated LPA flow that the disappearance of the false vacuum is not a truncation artifact.
minor comments (4)
  1. [Sec. 3.4, Eqs. (3.21)-(3.22)] The values quoted in Eq. (3.22) give u0 beta0^2 / m0^2 approximately 1.9, whereas Eq. (3.21) quotes approximately 0.32 / 0.22^2 = 6.6; please reconcile these numbers or clarify that they refer to different points in the acceptance region.
  2. [Sec. 2.1] The sentence 'the theory could loose its predictive power' should read 'lose'; similar typographical errors occur elsewhere in the manuscript.
  3. [Fig. 4] The color-coded acceptance regions in Fig. 4 will not be readable in grayscale; consider adding labels or hatching for the different confidence regions.
  4. [Sec. 3.5] The statement that for beta-hat ~ 300 'the scale of inflation exceeds the GUT scale by more than four orders of magnitude' appears to conflict with Eq. (3.20), where a larger beta-hat corresponds to a smaller tensor-to-scalar ratio and hence a smaller k_i; please clarify or correct this sentence.

Circularity Check

3 steps flagged · score 6.0 of 10

The RG-released false vacuum is identified with the fitted slow-roll start point, and the UV potential is its inverse RG image, so the mechanism's endpoint is imposed as the IR matching condition.

  1. fitted input called prediction [Sec. 3.1, Eq. (3.11)]
    "we fix the value of the running Fourier amplitude u_k and also the value of the constant dimensionful frequency β and mass m at the scale of inflation k_i, e.g., at the GUT scale, k_i∼k_GUT = 2×10^16 GeV, where one can match them with the slow-roll parameters (u_0, β_0, m_0). Once more, we emphasize that it is the scale of inflation k_i, i.e., the starting point of the slow-roll, not the Planck scale, where the parameters are matched, according to u_k_i = u_0, β = β_0, m = m_0."

    The endpoint of the proposed mechanism—the scale at which the false vacuum disappears and the VeV is released into slow-roll—is not obtained from the RG flow but is set by hand as the IR matching scale k_i. The potential at k_i is the slow-roll potential fitted to Planck data, so the field beginning to roll at k_i is the fit input, not a prediction. The UV false-vacuum potential is then produced by running the same fitted potential backward through Eq. (3.10); it is a re-expression of the IR fit plus the chosen β, and cannot independently confirm the release mechanism.

  2. self definitional [Sec. 3.3, after Eq. (3.18)]
    "Let φ_i denote the VeV of the value of the field φ at the onset of the slow-roll, i.e., the VeV of the potential at the very point in the RG analysis where the 'false' vacuum disappears and the slow-roll begins."

    This sentence defines the false-vacuum-disappearance point as the onset of slow-roll. Since the normalization of the potential at φ_i fixes u_0 through Eq. (3.19), the claimed RG-induced release is identical by construction to the slow-roll initial condition used in the Planck fit. The RG flow is not used to locate the disappearance of the false vacuum; the location is defined to be where the fitted slow-roll begins.

1 more flagged steps
  1. renaming known result [Sec. 4 (Conclusions)]
    "The method of RG-running induced inflation, as we show, determines a unique initial value as a starting point for the VeV, and so, it explains why the inflaton field starts in the particular slow roll domain. By a unique initial value as a starting point for the VeV, we mean that in the RG-induced inflationary scenario, the VeV, i.e., the false vacuum which is the initial value for the slow-roll depends only on the shape of the potential."

    The 'unique initial value' is the field value φ_i obtained from the slow-roll integral Eq. (3.16) with N in 50–60 using the fitted potential; it depends on the fitted shape of the potential in any slow-roll model. Calling it the false-vacuum value in the RG scenario renames the fitted initial condition as a prediction of the mechanism. The RG running does not independently determine φ_i; it only supplies a high-energy potential that, when run down to k_i, returns the fitted potential.

full rationale

The paper contains substantial independent components: the functional RG flow equation (3.1), the analytic solution (3.10), and the slow-roll fit to Planck data are all real calculations, and the MSG model's agreement with Planck in the large-β acceptance region is a non-trivial external check. No load-bearing self-citation chain was found; the cited external literature on k∼1/t and curved-space convexification is used as support but is not a circular uniqueness argument. However, the central narrative—that a false vacuum at the Planck scale is released by RG running to initiate inflation—reduces at its endpoint to the slow-roll fit. The scale k_i is defined as the starting point of slow-roll, and the potential at k_i is matched to the slow-roll parameters from the Planck fit; the false-vacuum disappearance is then defined as beginning at that same fitted point. The UV false-vacuum potential is obtained by inverting the same flow from these fitted values, so its existence is not an independent prediction. The 'unique initial value' advertised in the conclusions is the fitted slow-roll initial field value, renamed as a product of the RG mechanism. These are cases where a 'prediction' is equivalent to the matching condition imposed at the start, giving partial circularity rather than a fully self-contained derivation of the initial condition.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central mechanism rests on standard RG machinery plus four domain assumptions: the k-H identification, flat-space applicability, single-mode truncation, and an assumed false-vacuum initial condition. The model parameters are fitted to Planck data, so the agreement is not a parameter-free prediction.

free parameters (5)
  • mass m0 at inflation scale = small beta: 1.42e13 GeV; large beta: 1.3e12 GeV (5.4e-10 mp)
    Mass of the MSG potential at the inflation scale, fixed by the slow-roll fit to Planck data and the normalization condition Eq (3.19).
  • Fourier amplitude u0 at inflation scale = small beta: 2.4e64 GeV^4; large beta: 1.5e-21 mp^4
    Amplitude of the periodic term at the inflation scale, set by the normalization condition Eq (3.19) and the acceptance region in Fig 4.
  • frequency beta0 = small beta: 1.25e-19 GeV^-1 (beta_hat=0.3); large beta: 30/mp (beta_hat=30)
    Frequency of the periodic term, selected from Planck acceptance contours rather than derived; it controls the magnitude of the RG running through Eq (3.10).
  • e-fold number N = 55 (range 50 to 60 used)
    Chosen to define the initial field value via Eq (3.16); the acceptance regions and fitted parameters depend on this choice.
  • scale of inflation k_i = 2.0e13 GeV for large beta; 1.5e16 GeV for small beta
    Determined through Eq (3.20) from the fitted tensor-to-scalar ratio and normalization, so it is not an independent input.
assumptions (6)
  • standard math Wetterich flow equation in LPA with Litim regulator (Eq 3.1) describes the exact RG running of the MSG potential in d=4.
    Used to derive the running equations (Eq 3.6 to 3.7); this is the standard functional RG tool the paper adopts.
  • standard math The effective potential becomes convex in the IR (Eq C.12).
    Basis for the claim that false vacua disappear under RG flow; this follows from a Legendre-transform argument in Appendix C.
  • domain assumption The running RG scale k can be identified with the inverse cosmological time or Hubble scale (Appendix A).
    Heuristically assumed by citing Ref [10]; essential for interpreting pre-inflationary RG running as temporal evolution.
  • domain assumption Flat-space RG flow applies to the FLRW pre-inflationary background.
    The calculation uses the flat Euclidean action (Eq 2.5); curved-space effects are only cited via Fig 9 of Ref [14], not computed.
  • domain assumption Single-Fourier-mode truncation of the MSG flow (Eq 3.6) is sufficient.
    Higher harmonics are discarded after Fourier expansion of Eq (3.5); the paper does not estimate the error from this truncation.
  • ad hoc to paper At the Planck scale the field VeV is initially trapped in the false vacuum of the MSG potential.
    This is the initial condition the paper postulates; no dynamical mechanism populating the false vacuum is provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Renormalization-Group Running Induced Cosmic Inflation." pith.science (2026). https://pith.science/paper/BQWUPJ4G

@misc{pith2026190900580,
  author       = {Pith},
  title        = {Pith review of: Renormalization-Group Running Induced Cosmic Inflation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BQWUPJ4G}},
  note         = {Machine review of arXiv:1909.00580}
}
read the original abstract

As a contribution to a viable candidate for a standard model of cosmology, we here show that pre-inflationary quantum fluctuations can provide a scenario for the long-sought initial conditions for the inflaton field. Our proposal is based on the assumption that at very high energies (higher than the energy scale of inflation) the vacuum-expectation value (VeV) of the field is trapped in a false vacuum and then, due to renormalization-group (RG) running, the potential starts to flatten out toward low energy, eventually tending to a convex one which allows the field to roll down to the true vacuum. We argue that the proposed mechanism should apply to large classes of inflationary potentials with multiple concave regions. Our findings favor a particle physics origin of chaotic, large-field inflationary models as we eliminate the need for large field fluctuations at the GUT scale. In our analysis, we provide a specific example of such an inflationary potential, whose parameters can be tuned to reproduce the existing cosmological data with good accuracy.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 36 canonical work pages

  1. [14]

    Guilleux and J

    M. Guilleux and J. Serreau, Quantum scalar fields in de Sitter space from the nonperturbative renormalization group, Phys. Rev. D 92, 084010 (2015)

    work page 2015

  2. [1]

    A. H. Guth, Inflationary universe: A possible solution to the horizon and flatness problems , Phys. Rev. D 23, 347–356 (1981)

    work page 1981

  3. [2]

    A. A. Starobinsky, A new type of isotropic cosmological models without singularity , Phys. Lett. B 91, 99–102 (1980); V. F. Mukhanov and G. V. Chibisov, Quantum fluctuations and a nonsingular universe, JETP Lett. 33, 532–535 (1981) [Pisma Zh. Eksp. Teor. Fiz. 33, 549–553 (1981)]

    work page 1980

  4. [3]

    A. D. Linde, A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems , Phys. Lett. B 108, 389–393 (1982); A. Albrecht and P. J. Steinhardt, Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking, Phys. Rev. Lett. 48, 1220–1223 (1982)

    work page 1982

  5. [4]

    Martin, C

    J. Martin, C. Ringeval and V. Vennin, Encyclopaedia Inflationaris, Phys. Dark Univ. 5-6, 75–235 (2014)

    work page 2014

  6. [5]

    Friedmann, ¨Uber die Kr¨ ummung des Raumes, Z

    A. Friedmann, ¨Uber die Kr¨ ummung des Raumes, Z. Phys. A 10, 377–386 (1922); G. Lemaˆ ıtre, A Homogeneous Universe of Constant Mass and Increasing Radius accounting for the Radial Velocity of Extra-galactic Nebulae , Mon. Not. R. Astron. Soc. 91, 483–490 (1931); H. P. Robertson, Kinematics and World-Structure , Astrophys. J. 82, 284 (1935); A. G. Walker,...

    work page 1922

  7. [6]

    P. A. R. Ade et al. [Planck collaboration], Planck 2015 results. XIII. Cosmological parameters , Astron. Astrophys. 594, A13 (2016); P. A. R. Ade et al. [Planck collaboration], Planck 2015 results. XX. Constraints on inflation , Astron. Astrophys. 594, A20 (2016); P. A. R. Ade et al. [BICEP2 and Keck Array collaboration], Improved Constraints on Cosmology ...

    work page 2016

  8. [7]

    A. D. Linde, Chaotic inflation, Phys. Lett B 129, 177–181 (1983); Eternally existing self-reproducing chaotic inflanationary universe , Phys. Lett. B 175, 395–400 (1986)

    work page 1983

Show all 40 references
  1. [8]

    Earman and J

    J. Earman and J. Mosterin, A Critical Look at Inflationary Cosmology , Philosophy of Science. 66, 1–49 (1999); P. J. Steinhardt, The inflation debate , Scientific American 304, 36–43 (2011)

    1999

  2. [9]

    Rubio, Higgs Inflation, Front

    J. Rubio, Higgs Inflation, Front. Astron. Space Sci. 5, 50 (2019)

    2019

  3. [10]

    constant

    A. Bonanno and M. Reuter, Cosmology of the Planck era from a renormalization group for quantum gravity, Phys. Rev. D 65, 043508 (2002); A. Bonanno and M. Reuter, Cosmology with self-adjusting vacuum energy density from a renormalization group fixed point , Phys. Lett. B 527, 9–...

    2002

  4. [11]

    Cai, Y.-C

    Y.-F. Cai, Y.-C. Chang, P. Chen, D. A. Easson and T. Qiu, Planck constraints on Higgs modulated reheating of renormalization group improved inflation , Phys. Rev. D 88, 083508 (2013); G. Kofinas and V. Zarikas, Asymptotically safe gravity and nonsingular inflationary big – 29 – b...

    2013

  5. [12]

    Kaya, Exact renormalization group flow in an expanding Universe and screening of the cosmological constant, Phys

    A. Kaya, Exact renormalization group flow in an expanding Universe and screening of the cosmological constant, Phys. Rev. D 87, 123501 (2013)

    2013

  6. [13]

    Serreau, Renormalization group flow and symmetry restoration in de Sitter space , Phys

    J. Serreau, Renormalization group flow and symmetry restoration in de Sitter space , Phys. Lett. B 730, 271–274 (2014); T. Prokopec and G. Rigopoulos, Functional renormalization group for stochastic inflation , J. Cosmology Astropart. Phys. 1808, 013 (2018)

    2014

  7. [15]

    Guberina, R

    B. Guberina, R. Horvat, and H. Stefancic, Renormalization-group running of the cosmological constant and the fate of the universe , Phys. Rev. D 67, 083001 (2003); M. Reuter and H. Weyer, Renormalization group improved gravitational actions: A Brans-Dicke approach , Phys. Rev....

    2003 arXiv

  8. [16]

    Sola, Dark energy: a quantum fossil from the inflationary Universe? , J

    J. Sola, Dark energy: a quantum fossil from the inflationary Universe? , J. Phys. A 41, 164066 (2008); J. Sola, A. Gomez-Valent, J. De Cruz Perez, First Evidence of Running Cosmic Vacuum: Challenging the Concordance Model , Astrophys. J. 836, 43 (2017); S. Basilakos, N. E. Mavr...

    2008

  9. [17]

    J. Sola, A. Gomez-Valent, The ΛCDM Cosmology: From inflation to dark energy through running Λ, Int. J. Mod. Phys. D 24, 1541003 (2015)

    2015

  10. [18]

    Basilakos, N

    S. Basilakos, N. E. Mavromatos, J. Sola Paracaula, Gravitational and Chiral Anomalies in the Running Vacuum Universe and Matter-Antimatter Asymmetry , Phys. Rev. D 101, 045001 (2020)

    2020

  11. [19]

    L. P. Kadanoff, Scaling laws for ising models near Tc, Physics 2, 263–272 (1966)

    1966

  12. [20]

    K. G. Wilson, Model Hamiltonians for Local Quantum Field Theory , Phys. Rev. 140, B445–B457 (1965) – 30 –

    1965

  13. [21]

    Polonyi, Lectures on the functional renormalization group method , Cent

    J. Polonyi, Lectures on the functional renormalization group method , Cent. Eur. J. Phys. 1, 1–71 (2003)

    2003

  14. [22]

    Baumann, TASI Lectures on Inflation, available as arXiv:0907.5424 [hep-th]

    D. Baumann, TASI Lectures on Inflation, available as arXiv:0907.5424 [hep-th]

    arXiv

  15. [23]

    Freese, J

    K. Freese, J. A. Frieman and A. V. Olinto, Natural inflation with pseudo Nambu-Goldstone bosons, Phys. Rev. Lett. 65, 3233–3236 (1990); K. Freese and W. H. Kinney, Natural inflation: consistency with cosmic microwave background observations of Planck and BICEP2 , J. Cosmology As...

    1990

  16. [24]

    Kobayashi, O

    T. Kobayashi, O. Seto and Y. Yamaguchi, Axion monodromy inflation with sinusoidal corrections, Prog. Theor. Exp. Phys. 2014, 103E01 (2014); T. Higaki, T. Kobayashi, O. Seto and Y. Yamaguchi, Axion monodromy inflation with multi-natural modulations , J. Cosmology Astropart. Phys....

    2014

  17. [25]

    Wetterich, Exact evolution equation for the effective potential , Phys

    C. Wetterich, Exact evolution equation for the effective potential , Phys. Lett. B 301, 90–94 (1993); T. R. Morris, The Exact renormalization group and approximate solutions , Int. J. Mod. Phys. A 9, 2411–2449 (1994)

    1993

  18. [26]

    L. V. Keldysh, Diagram technique for nonequilibrium processes, Zh. Eksp. Teor. Fiz. 47, 1515–1527 (1964); [Sov. Phys. JETP 20, 1018 (1965)]; O. V. Konstantinov and V. I. Perel, A graphical technique for computation of kinetic quantities , Zh. Eksp. Teor. Fiz. 39, 197 (1960); [...

    1964

  19. [27]

    Polonyi, Quantum-classical crossover in electrodynamics, Phys

    J. Polonyi, Quantum-classical crossover in electrodynamics, Phys. Rev. D 74, 065014 (2006); S. Nagy, J. Polonyi, I. Steib, Quantum renormalization group, Phys. Rev. D 93, 025008 (2016); S. Nagy, J. Polonyi, I. Steib, Euclidean scalar field theory in the bilocal approximation , ...

    2006

  20. [28]

    Brustein, S

    R. Brustein, S. P. de Alwis and P. Martens, Cosmological stabilization of moduli with steep potentials, Phys. Rev. D 70, 126012 (2004)

    2004

  21. [29]

    Codello, N

    A. Codello, N. Defenu and G. D’Odorico, Critical exponents of O(N) models in fractional dimension, Phys. Rev. D 91, 105003 (2015)

    2015

  22. [30]

    Defenu and A

    N. Defenu and A. Codello, Scaling solutions in the derivative expansion , Phys. Rev. D 98, 016013 (2018)

    2018

  23. [31]

    D. F. Litim, Optimisation of the exact renormalisation group , Phys. Lett. B 486, 92–99 (2000)

    2000

  24. [32]

    Nandori, S

    I. Nandori, S. Nagy, K. Sailer and A. Trombettoni, Comparison of renormalization group schemes for sine-Gordon-type models , Phys. Rev. D 80, 025008 (2009); I. Nandori, On the renormalization of the bosonized multi-flavor Schwinger model , Phys. Lett. B 662, 302–308 (2008); S. ...

    2009

  25. [33]

    Nandori, Bosonization and functional renormalization group approach in the framework of QED2, Phys

    I. Nandori, Bosonization and functional renormalization group approach in the framework of QED2, Phys. Rev. D 84, 065024 (2011)

    2011

  26. [34]

    Gies, Introduction to the Functional RG and Applications to Gauge Theories , in Springer Notes in Physics vol

    H. Gies, Introduction to the Functional RG and Applications to Gauge Theories , in Springer Notes in Physics vol. 852 , J. Polonyi and A. Schwenk (Eds.), pp. 287–348 Springer (Heidelberg), 2012. See also e-print hep-ph/0611146

    2012 arXiv

  27. [35]

    Delamotte, An Introduction to the Nonperturbative Renormalization Group , in Springer Notes in Physics vol

    B. Delamotte, An Introduction to the Nonperturbative Renormalization Group , in Springer Notes in Physics vol. 852 , J. Polonyi and A. Schwenk (Eds.), pp. 49–85, Springer (Heidelberg),

  28. [36]

    D. H. Lyth, Particle Physics Models of Inflation , Lect. Notes Phys. 738 81-118 (2008). – 31 –

    2008

  29. [37]

    D. H. Lyth and A. Riotto, Particle physics models of inflation and the cosmological density perturbation, Phys. Rept. 314 1-146 (1999)

    1999

  30. [38]

    R. J. Rivers, Path Integral Methods in Quantum Field Theory , (Cambridge University Press, 1987)

    1987

  31. [39]

    U. D. Jentschura, N. Defenu, I. G. M´ ari´ an, I. N´ andori, and A. Trombettoni,Role of Field-Independent Terms in Nonperturbative RG , in preparation (2019). – 32 –

    2019

  32. [2012]

    See also e-print cond-mat/0702365

    arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.