Standard Running, "Physical Running", Cosmological Constant and Newton Coupling

Hikaru Kawai; Nobuyoshi Ohta

REVIEW 3 major objections 4 minor 2 cited by

Standard Running, "Physical Running", Cosmological Constant and Newton Coupling

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Standard beta functions are not wrong when the renormalization scale is chosen well, and vacuum energy and Newton coupling do run once spacetime is curved.

desk verdict A clean restatement of the standard-"physical" running equivalence in Section 2, followed by a schematic curved-spacetime argument whose advertised conclusion is undermined by the paper's own wave-function renormalization caveat. read the letter →

arxiv 2505.16578 v3 pith:FPQ2XKFJ submitted 2025-05-22 hep-th gr-qc

Hikaru Kawai , Nobuyoshi Ohta This is my paper

classification hep-thgr-qc

keywords renormalizationgroupbetafunctionrunningcouplingpointcosmologicalconstantNewtonvacuumenergycurvedspacetime

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 defends the standard renormalization-group $\beta$ function against recent claims that it gives the wrong “physical running” of coupling constants. It shows the apparent failure comes from fixing the renormalization point at $\mu=p$, and that a suitable choice of $\mu$ removes the large logarithms and reproduces the same momentum dependence as the proposed alternative. The paper then argues that the vacuum energy and the Newton coupling, usually said not to run, acquire a scale dependence in curved spacetime because the curvature radius $L$ supplies a physical scale. The authors caution that only wave-function-renormalization-independent combinations, such as $\eta = 16\pi G\sqrt{\rho}$, have an unambiguous running.

What carries the argument

The load-bearing object is the renormalization-point ($\mu$) independence of physical amplitudes. In the flat-space argument, the standard scheme keeps $\lambda$ and chooses $\mu$ to absorb the large logarithm, while the “physical running” scheme redefines the coupling and sets $\mu=p$; the two are shown to produce the same $1/\lambda$ as a function of $p^2$. For the gravitational part, the central mechanism is the assumed one-loop divergence structure Eq. (23), $\varepsilon_{\rm 1-loop} = a\Lambda^4 + (b m^2 + c/L^2)\Lambda^2 + (d_1 m^4 + d_2 m^2/L^2 + d_3/L^4)\log\bigl(f(m^2,1/L^2)/\Lambda^2\bigr)$, which provides the scale $f(m^2,1/L^2)$ that makes the vacuum energy and Newton coupling run with $L$ through the RG equations (27) and solutions (28)–(30).

What would settle it

Compute the exact one-loop effective action for a massive scalar on a specific maximally symmetric background such as de Sitter space with radius $L$. If the coefficient of $m^2 \log(f/\mu^2)$ in the renormalized $1/16\pi G$ vanishes or $f$ has no $L$-dependence, the claimed running of the Newton coupling would be absent. For the $\beta$-function part, an explicit two-loop or exact amplitude calculation in a model with the structure of Eq. (1) could settle whether standard RG with optimized $\mu$ reproduces the full amplitude; if standard RG fails while the redefined coupling succeeds, the equivalence claim would be falsified.

Watch

Extended reading notes

Core claim

In flat spacetime, an amplitude with logarithms involving a mass $m$ and momentum $p$ yields the standard $\beta$ function $\beta_\lambda = 2(a+b)\lambda^2$ from the requirement that physical quantities be independent of $\mu$. The “physical running” prescription redefines the coupling as $\lambda' = \lambda + (a-c)\lambda^2 \log(m^2/\mu^2)$ and obtains $\beta_{\lambda'} = 2(b+c)\lambda'^2$; the paper shows both routes agree once $\mu$ is chosen to remove the large logarithm, so the disagreement is an artifact of assuming $\mu \approx p$. For gravity, the paper considers the one-loop vacuum energy on a curved background with curvature radius $L$, where the divergences take the form of Eq. (23). Renormalizing the $R^2$, Newton, and vacuum-energy terms gives RG equations whose solutions, Eq. (28), become $L$-dependent after choosing $\mu^2 = f(m^2, 1/L^2)$, Eq. (30). The paper therefore concludes that the vacuum energy and Newton coupling do run in curved spacetime.

Load-bearing premise

The curved-spacetime running claim depends on the assumed one-loop divergence formula (23), whose coefficients and the mass-curvature function $f$ are not explicitly computed; if that formula differs for an actual spacetime, the claimed $L$-dependence of the vacuum energy and Newton coupling would change.

Editorial extensions

If this is right

The standard beta function and the “physical running” beta function describe the same physics; apparent disagreements signal an inconvenient choice of renormalization point, not a failure of the standard RG.
Existing flat-space RG-improved calculations remain valid as long as $\mu$ is chosen to suppress large logarithms, so the recent critique does not require abandoning standard running couplings.
In curved spacetime, the vacuum energy and the Newton coupling depend on the curvature radius $L$, so claims that these constants do not run must be qualified by the background geometry.
Only wave-function-renormalization-independent quantities such as $\eta = 16\pi G\sqrt{\rho}$ have physical meaning as running couplings; the separate flows of $\rho$ and $G$ are scheme-dependent.

Reading between the lines

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

Editorial — One could test the flat-space equivalence at two loops: if a two-loop calculation in a model with the structure of Eq. (1) shows that standard RG with optimized $\mu$ fails to reproduce the exact amplitude while the redefined coupling succeeds, the paper's equivalence claim would need modification.
Editorial — If Eq. (23) is correct, the same curvature-dependent logarithms should appear in the heat-kernel coefficients of any maximally symmetric background, giving a concrete prediction that can be checked by computing the one-loop effective action on de Sitter space.
Editorial — The wave-function-renormalization caveat suggests that quantum-gravity studies should formulate their flow equations in terms of field-redefinition-invariant combinations such as $\eta$, which could reconcile conflicting claims about whether the cosmological constant runs.

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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

Section 2 is the strongest part of the paper. It takes the Buccio-Donoghue-Percacci examples and shows in detail that the "physical" beta function is just the standard beta function plus a finite coupling redefinition, and that the appearance of large logarithms in the standard scheme comes from forcing mu = p when another choice removes them. This is not a new result, but it is clearly explained and should settle a fairly pointless controversy if people actually read it. The RG equivalence is shown to O(lambda^2), which covers the cited examples.

Section 3 is where my enthusiasm drops. The idea that a curved background with curvature radius L gives a physical scale so that vacuum energy and Newton coupling can run is plausible and worth discussing. But the execution is schematic. Eq. (23) simply asserts the one-loop divergence structure with coefficients d_i and a function f left unspecified. Without those, Eqs. (27) and the solutions (28)/(30) are not derivations; they are templates. More importantly, the stress-test note is right: the separate running of rho and G is not invariant under wave-function renormalization. A metric rescaling changes both, and the paper's own final paragraph concedes that only eta = 16 pi G sqrt(rho) is physically meaningful. That means the abstract's claim that the vacuum energy and the Newton coupling run is, as stated, not well-defined unless a normalization convention is fixed and the eta flow is computed. The paper does not do that. So the central advertised claim is not established; what is established is that a particular choice of normalization could produce such running, and that eta is a candidate invariant that might run.

The paper is honest about this limitation, which I credit. But the abstract and introduction overstate the conclusion. If the authors fixed a metric normalization and computed the eta beta function, or at least showed how the separate flows combine into an invariant, Section 3 would be a real result. As it stands, it is a plausible argument with a missing load-bearing step.

Recommendation: send to a referee. Section 2 alone is worth publishing as a clarification, and Section 3 raises a question that the asymptotically safe gravity community should confront. The referee should push for either a derivation of the eta flow or a revised abstract that aligns with the caveat. For my own work, I would not cite this for the running claim, but I might cite Section 2 as a clean statement of scheme equivalence.

Referee Report

3 major / 4 minor

Summary. The paper has two parts. Section 2 uses a toy amplitude of the form (1) to argue that the recent proposal of a "physical running" coupling in Refs. [4-6] does not show that the standard beta function is wrong. The authors demonstrate that the standard beta function (3) and the modified beta function (7) are equivalent through the coupling redefinition (5), and that the apparent failure of the standard result comes from the special choice mu=p. Choosing mu to eliminate large logarithms, as in (10), reproduces the correct momentum dependence. Section 3 attempts to extend this logic to the vacuum energy and Newton coupling. On a curved background with curvature radius L, the authors assume a one-loop divergence structure (23), write the RG equations (27), solve them in (28), and choose mu^2=f(m^2,1/L^2) in (29) to obtain the L-dependent expressions (30). The paper concludes that the vacuum energy and Newton coupling run in curved spacetime, but adds a final caveat that only the combination eta=16 pi G sqrt(rho) is invariant under wave-function renormalization and that separate RG flows of rho and G are not meaningful.

Significance. Section 2 is a clean and useful clarification of a current controversy: it shows explicitly that the standard and "physical" beta functions are related by a finite redefinition and that the choice of renormalization point is a matter of convenience. This part is likely correct and could be pedagogically valuable. Section 3 addresses a timely question about whether the cosmological constant and Newton coupling run, and the idea of using a curved background to provide a physical scale is interesting. However, the central claim of Section 3 is not yet fully established. The assumed divergence structure in Eq. (23) is not derived, and more importantly, the separate running of rho and G is not invariant under wave-function renormalization, as the authors themselves concede in the final paragraph. The paper is not circular: the self-citations [14,15] appear only in the caveat, and no equation is constructed to force the final result. With a fixed normalization or a derivation of the flow of the observable combination eta, the Section 3 claim could become solid; as written, Eq. (30) is a convention-dependent statement.

major comments (3)

[Section 3, Eqs. (27)-(30) and final paragraph] The advertised running of rho and G is not well-defined because the action (17) is invariant under the simultaneous wave-function renormalization g->Z g with rho->Z^2 rho and 1/(16 pi G)->Z/(16 pi G). The RG equations (27) do not fix Z, so one can choose Z(mu) to make rho constant along the flow, changing the apparent running of G. The final paragraph correctly states that only eta=16 pi G sqrt(rho) is independent of this rescaling, but the paper does not derive the flow of eta. Therefore the conclusion "the vacuum energy and the Newton coupling run" is not established as a physical statement; it holds only in a particular normalization that is left unspecified.
[Section 3, Eq. (23)] The one-loop divergence structure in Eq. (23) is assumed, with the constants d1, d2, d3 and the homogeneous function f left unspecified. The running equations (27) and the L-dependence of (30) depend on d1, d2, and the explicit form of f. Without a derivation of these coefficients, even the sign and existence of the claimed running are not established. At minimum, the flat-space limit L->infinity should reproduce the logarithm in Eq. (16), which would determine d1; this consistency check is not provided.
[Section 3, paragraph after Eq. (24)] The paper states that the d3 L^{-4} log(mu^2/ Lambda^2) term is removed by renormalization of R^2 and R_{mu nu}^2 terms, but the starting action (17) contains no such curvature-squared terms. If these terms are generated at one loop, they should be included in the effective action for consistency; if they are not, the cancellation of the d3 term requires a different explanation. This point does not affect the running of rho and G directly, but it makes the renormalization prescription in Eqs. (25)-(27) less transparent.

minor comments (4)

[Eq. (22)] The expression for epsilon[g] is hard to parse because of the typesetting "1R d^4x sqrt(-g)"; it should be written as 1/(Integral d^4x sqrt(-g)).
[Eq. (23), text following] The phrase "homogeneous function of degree 1" is ambiguous; f(m^2,1/L^2) has mass dimension two, and the intended meaning is that f is homogeneous of degree one in the two arguments m^2 and L^{-2}.
[Section 2, Eqs. (1)-(14)] The notation in Eq. (1) suppresses the second-order terms in the RG equation (2); writing the higher-order terms explicitly or stating that they are O(lambda^3) would improve readability.
[Abstract and Section 3, final paragraph] The abstract claims that the vacuum energy and Newton coupling run, but the final paragraph states that only the combination eta=16 pi G sqrt(rho) is meaningful. This tension should be resolved, either by softening the abstract or by deriving the flow of eta.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the scale-choice argument and RG equations are self-contained; the only self-citation appears in the final wave-function caveat and is not load-bearing for the main derivation.

full rationale

Section 2 is an internal consistency check: Eq. (3) follows from µ-independence of M(p), and Eq. (7) is derived from Eq. (3) after the redefinition (5), so the standard and 'physical' beta functions are shown equivalent rather than one being defined as the other. Section 3 starts from an assumed one-loop divergence structure, Eq. (23); Eqs. (27) are the corresponding RG equations, and Eq. (30) is the standard RG improvement obtained by choosing µ² = f(m²,1/L²) to remove large logarithms in Eq. (25). This is a conventional scale choice, not a fit of the claimed L-dependence, and the unspecified coefficients d_i affect the magnitude of the effect but do not make the derivation circular. The paper itself flags the one real limitation: in the final paragraph it concedes, citing its own Refs. [14,15], that 'it does not make sense to discuss the RG flow for the vacuum energy and Newton coupling separately' because wave-function renormalization can shift them, and identifies η ≡ 16πG√ρ as the invariant that runs. That limitation is weighed: Eqs. (27)-(30) give a convention-dependent split of ρ and G, so the advertised 'both run' statement should be read as the η-flow; this is a correctness and interpretation concern rather than a self-referential reduction. No equation in the paper is constructed to equal the target result, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard RG machinery plus an assumed heat-kernel divergence structure in curved spacetime. No fundamentally new entities are introduced, but the unspecified coefficients and function f constitute free parameters, and the identification of mu with the curvature scale is an added modeling choice.

free parameters (3)

integration constants rho0 and G0
Boundary conditions for the RG solutions in Eq. (28); they set the renormalized values at a reference scale but are not fitted to data.
coefficients of f(m^2,1/L^2) = c1, c2 (unspecified)
The shape-dependent homogeneous function f is left as c1 m^2 + c2/L^2 with c1, c2 undetermined; the running in Eqs. (29)-(30) depends on this choice.
divergence coefficients d1, d2, d3
The coefficients in Eq. (23) are not computed, so the running equations (27) are symbolic rather than concrete predictions.

assumptions (4)

domain assumption The one-loop divergences in curved spacetime take the form of Eq. (23) with L as the only additional scale.
Invoked in Section 3; no explicit heat-kernel calculation is provided.
domain assumption The background is maximally symmetric so that <T_mu_nu> = -epsilon g_mu_nu.
Used to define the vacuum energy in Eq. (21).
ad hoc to paper mu can be identified with the physical scale f(m^2,1/L^2) to define running.
The choice mu^2 = f(m^2,1/L^2) in Eq. (29) is what turns the mu-dependence into L-dependence; it is a conventional but not mandatory identification.
standard math Dimensional regularization and MS scheme are valid in curved spacetime.
Assumed throughout Section 3.

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Cite this review

Pith. "Pith review of Standard Running, "Physical Running", Cosmological Constant and Newton Coupling." pith.science (2026). https://pith.science/paper/FPQ2XKFJ

@misc{pith2026250516578,
  author       = {Pith},
  title        = {Pith review of: Standard Running, "Physical Running", Cosmological Constant and Newton Coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FPQ2XKFJ}},
  note         = {Machine review of arXiv:2505.16578}
}

abstract

Recently it is asserted that the standard beta function does not describe the correct running of the coupling constant in some theories. We show that the problem arises from the assumption $\mu=p$ ($\mu$ is a renormalization point) and that a suitable choice of $\mu$ gives the correct running. It is also claimed that neither the cosmological constant nor Newton coupling run. We argue that running can be discussed when we consider the curved spacetime.

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Forward citations

Cited by 2 Pith papers

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

Scaling solutions for gauge invariant flow equations in dilaton quantum gravity
hep-th 2025-12 conditional novelty 6.0 of 10

Scaling solutions of a gauge-invariant functional flow equation support the dilaton quantum gravity fixed point, with Planck mass ~ φ² at large field and a stable negative kinetial in the infrared.
Running Vacuum in the expanding Universe: a unified QFT paradigm for Inflation and Dark Energy
gr-qc 2026-06 unverdicted novelty 3.0 of 10

The running vacuum model derives dynamical vacuum energy from QFT in curved spacetime, using H^4 terms for inflation and H^2 terms for dark energy while G evolves logarithmically.

Reference graph

Works this paper leans on

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