On the scaling behavior of the cosmological constant and the possible existence of new forces and new light degrees of freedom

Ilya Shapiro; Joan Sola

REVIEW

On the scaling behavior of the cosmological constant and the possible existence of new forces and new light degrees of freedom

Ilya Shapiro, Joan Sola

High Energy Physics - Phenomenology · Submitted 25 Oct 1999 · 24 references

Pith's verdict: The one-loop cancellation in Eq. (9) is real and worth knowing; the light-scalar fix for the sign is where the paper is soft, and the stress-test's threshold objection lands.

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CitationIlya Shapiro, Joan Sola. "On the scaling behavior of the cosmological constant and the possible existence of new forces and new light degrees of freedom." arXiv:hep-ph/9910462 (1999). https://pith.science/paper/AKEKESBK

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The core of this paper holds up: the one-loop cancellation in Eq. (9) is real, and the resulting running of the physical CC in Eq. (10) is a clean piece of SM bookkeeping that I don't think was written down in this form before. The leading m^4 h^2/lambda terms in the running of Lambda_ind cancel against the vacuum term, leaving (4pi)^2 dLambda_ph/dt = -4 sum m_j^4. That is the paper's actual contribution, and it is worth knowing. The neutrino masses come from oscillation data, so the magnitude estimate is not fitted to the CC; that is the strongest non-circular element here. The authors also state the sign problem plainly rather than burying it, which I appreciate. The magnitude estimate itself is loose: integrating over the light neutrino range gives |Lambda_ph| ~ 10^-48 GeV^4, but only if you sit in the maximal-mixing MSW region; at small mixing the authors say it comes out about an order of magnitude low. The sign fix is the softest part, and the stress-test note is right about it. The scalar decouples below m_S, so its contribution at mu = m_e is 12 m_S^4/(4pi)^2 ln(m_e/m_S), not an integral from m_nu to m_S with the scalar active throughout. At m_S = 4 m_nu that gives roughly 350 m_nu^4, about two orders above the observed value; even the paper's own formula gives ~27 m_nu^4, about 20 times too big. To land in range you need m_S ~ m_nu to within a few percent, not 'a few times larger.' The tuning is tighter than the paper admits. This does not touch Eqs. (9) and (10), but it weakens the only proposed cure for the sign. The other soft spot is the IR boundary condition Lambda_ph(IR) = 0, which is assumed rather than derived; the authors acknowledge this in Section 5, so it is a known limitation, not a hidden one. The EFT freezing of couplings at the Fermi scale is also acknowledged and is a reasonable approximation at these energies. Who should read it: anyone working on RG running of the cosmological constant, neutrino-dark energy connections, or submillimeter gravity tests. The alpha = 1/3 prediction for a gravity-strength submillimeter force is a concrete target. Recommendation: send it out. The core calculation is sound and the speculative extension is clearly labeled. A referee should press on the threshold integration and the numerical estimate, but the paper deserves the review.

Referee report

Significance

The structural observation in Eq. (9) is genuine and useful: the leading m^4 h^2/lambda terms cancel between the induced and vacuum contributions, and the residual running is controlled by the lightest fermion masses. The algebra of Eqs. (9) and (10) checks out, and the connection to neutrino masses gives the magnitude estimate external grounding rather than a fit to the cosmological constant. The proposal of a light scalar mediating a submillimeter force is falsifiable and is the kind of concrete prediction that makes the framework worth pursuing. The paper is honest about the sign discrepancy and about the assumed infrared boundary condition. The main weakness is that the sign-fixing mechanism is not consistently implemented at the level of thresholds, which is load-bearing for the paper's central new-physics proposal.

Major objections (4)

[Section 4, Eq. (11) and the sentence 'm_S >~ 4 m_nu'] The threshold integration in Eq. (11) is inconsistent with the decoupling prescription stated in Section 2 ('at sufficiently small energies one can invoke the decoupling of the heavier degrees of freedom'). A scalar of mass m_S is active only for mu > m_S, so its contribution to Lambda_ph at mu = m_nu is 12 m_S^4/(4 pi)^2 ln(m_nu/m_S), which is negative for m_S > m_nu; it cannot flip the sign of Lambda_ph at the neutrino scale. The paper's integration 'in the interval from m_nu to m_S' treats the scalar as active below its own mass, which is backwards. Even taking Eq. (11) at face value, m_S = 4 m_nu gives 12 m_S^4/(4 pi)^2 ln 4 - 4 m_nu^4/(4 pi)^2 ln 4 ~ 27 m_nu^4, i.e. about 4 x 10^-46 GeV^4 for m_nu ~ 2 x 10^-3 eV, roughly twenty times the observed value, not 'the right ballpark'. To land in the observed range while keeping beta_Lambda > 0, m_S must be at or below m_nu, tuned at the few-percent level. This is load-bearing because the scalar is the paper's only proposed resolution of the sign problem; the tuning is stronger than the manuscript acknowledges.
[Section 2, Eq. (6)] Eq. (6) is stated for mu > M_F and is described as gaining contributions from all massive fields, but it lists only the Higgs doublet and the fermions. The W and Z bosons are massive in this domain and contribute to the vacuum bubble; their masses are proportional to g v, so their contributions to Lambda_v scale as g^4 m^4/lambda^2 and their running mixes g and lambda. The cancellation demonstrated in Eq. (9) uses the specific Higgs relation m_j = h_j v/sqrt(2) and lambda = 2 m^2/v^2, and it is not shown that the same cancellation survives in the gauge sector. Since the abstract claims that the running 'inevitably' produces a scaling dependence, the gauge-sector contribution should be computed explicitly and the cancellation checked, or the claim should be restricted to the Higgs/fermion sector.
[Section 3, Eq. (10) and the magnitude estimate] The integrated result scales as ln(m_nu_s/m_nu_e), which is not fixed by the oscillation data cited. For the quasi-degenerate double-hierarchy Ansatz m_nu_e ~ m_nu_s, the logarithm is small and uncertain, and the quoted |Lambda_ph| ~ 10^-48 GeV^4 depends sensitively on the assumed ratio (the paper's own example uses a factor 3/2). The abstract's statement that the order of magnitude is 'roughly the one derived from recent supernovae observations' therefore rests on a quantity that the cited data do not determine. The sensitivity of the result to the mass ratio should be stated explicitly, and the magnitude claim should be presented as a range rather than a prediction.
[Abstract and Section 5] The infrared boundary condition Lambda_ph(IR) = 0 is assumed rather than derived; the authors acknowledge this in Section 5 ('we take it as a natural fact based on the observed smallness of the measured CC'). Given that acknowledgment, the abstract's claim that the magnitude is 'roughly the one derived from recent supernovae observations' overstates what is a consistency statement contingent on the choice of the far-infrared scale mu_0. The abstract and the discussion should make clear that the result is a conditional consistency check, not a parameter-free prediction, and should state how the answer depends on the choice of mu_0.

Minor objections (6)

[Section 4, Eq. (16)] The relation m_S^2 = Lambda_QCD^4/M^2 is a dimensional estimate; the coefficient and the sign of m_S^2 are not computed. Please state explicitly that this is an order-of-magnitude estimate and discuss whether the sign is guaranteed to be positive.
[Section 4, Eq. (17) and the surrounding discussion] The identification of the dilatation-symmetry-breaking scale M with the right-handed Majorana mass scale is an additional hypothesis, not a consequence of the framework. The paper says the issue 'deserves further study'; it would help to state clearly at the point of Eq. (17) that this is a separate assumption.
[Section 4, the comparison with submillimeter bounds] The paper asserts that the new force with alpha = 1/3 and range 0.02-0.2 mm is not in contradiction with existing macroscopic experiments, citing Ref. [14]. A quantitative comparison with the bounds at the relevant Compton wavelength would make this claim checkable; at alpha = 1/3 the margin may be small.
[References] Ref. [5] is listed as 'work in progress', yet the running of G and the evolution of the cosmological constant at the Fermi epoch are deferred to it. Since these are part of the claimed consistency of the framework, the relevant statements should either be supported within this manuscript or explicitly flagged as unpublished.
[Section 3, the double-hierarchy Ansatz] The scenario relies on a sterile neutrino, which is not part of the Standard Model, and on a specific interpretation of the LSND and solar data. The model dependence of this choice should be stated more explicitly, and the alternative that the lightest active neutrino alone sets the scale should be mentioned.
[Manuscript throughout] The text contains numerous broken words and spacing artifacts (e.g. 'fre edom' in the title, 'Stan dard', 't he', 'renor malizability'). These should be cleaned up before publication.

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The paper

The paper claims

The observed cosmological constant may be the residue of Standard Model running between the lightest neutrino masses, with a wrong sign that hints at a new light scalar.

The reading

The Standard Model's Higgs mechanism contributes a vacuum energy some 55 orders of magnitude larger than the observed cosmological constant, so the paper starts from the standard assumption that a bare vacuum term cancels it exactly at some very far infrared scale. The question the authors ask is whether that cancellation survives when every ingredient — the Higgs mass, the couplings, and the vacuum term itself — is run according to the renormalization group. They show that the leading large terms cancel automatically in the Standard Model, leaving a residual drift of the physical cosmological constant proportional to the fourth powers of the fermion masses, so that integrating across the light neutrino thresholds produces a value of order $10^{-47}$ GeV$^4$, numerically in the range the supernovae data favour. The catch is that the sign comes out negative, opposite to what those data suggest. The authors propose that an extra very light scalar, a Cosmon-like dilaton with mass a few times the lightest neutrino mass, can flip the sign and, because of its gravitational-strength coupling to the anomalous trace of the energy-momentum tensor, would show up as a new force in the submillimetre range.

What carries the argument

The load-bearing object is the one-loop renormalization group equation for the physical cosmological constant in the far infrared, $(4π)^2 dΛ_{ph}/dt = 12 m_S^4 - 4Σ_i m_{ν_i}^4$, together with the cancellation that makes it small: because the induced term is $Λ_{ind} = -m^4/2λ$, its $\beta$ function built from the Standard Model equations for $m^2$ and $λ$ reproduces, at leading order, exactly the negative of the neutrino vacuum-bubble contribution. What survives is a fourth-power mass sum, so the size of the generated constant is set by the fourth powers of the lightest masses rather than by the electroweak scale. The second piece of machinery is the Cosmon: a non-linearly realized dilaton $S$ entering as $e^{S/M}$ multipliers, whose mass $m_S^2 = Λ_{QCD}^4/M^2$ comes from the QCD trace anomaly and the scale $M$ of dilatation breaking, giving $m_S ∼ 10^{-3}$ eV and a Compton wavelength of a fraction of a millimetre for $M ∼ 10^{10}$ GeV.

Extended reading notes

Core claim

The paper establishes that the fine-tuned cancellation between the Higgs-induced vacuum energy and the bare vacuum term is not stable under renormalization group flow, but that the breakdown is far gentler than one might fear. Working below the electron mass, where only neutrinos remain active, the authors show that the leading $m^4 h_j^2/λ$ contributions to the running of the induced term cancel against the corresponding vacuum-bubble contributions, leaving $dΛ_{ind}/dt = -(2/(4π)^2)Σ_j m_j^4$ and hence $(4π)^2 dΛ_{ph}/dt = -4Σ_j m_j^4$. Integrating this across the mass splittings of a double-hierarchy neutrino spectrum with $m_{ν_e} ≃ m_{ν_s} ≪ m_{ν_μ} ≃ m_{ν_τ}$ yields $|Λ_{ph}|$ in the range singled out by the supernovae and CMB data, roughly $(2 ± 1) × 10^{-47}$ GeV$^4$. The prediction's sign, however, is negative, so the authors add a light real scalar whose positive contribution $12 m_S^4$ to the $\beta$ function outweighs the neutrino term when $m_S ≳ 4 m_ν$, and they argue that a pseudo-Goldstone boson of spontaneously broken dilatation symmetry naturally has both the required mass and an almost decoupled phenomenology.

Load-bearing premise

The whole construction starts from the premise that the physical cosmological constant is exactly zero at some very far infrared scale, a fine-tuned boundary condition the paper explicitly declines to derive and instead takes as natural because the measured value is so small.

Editorial extensions

If this is right

The magnitude of the cosmological constant becomes a function of neutrino masses, so a measured absolute neutrino mass scale would translate directly into a predicted vacuum energy density.
A new scalar of mass a few times $10^{-3}$ eV should exist, producing a Yukawa-like force of roughly gravitational strength with a range of order $0.02$ to $0.2$ mm.
Because the coupling is to the anomalous trace of the energy-momentum tensor rather than the full trace, the new force carries composition-dependent baryon-number and isospin components in addition to the dominant mass-dependent term.
The intermediate scale $M ∼ 10^{10}$ GeV required for the Cosmon mass coincides with the scale needed for right-handed Majorana neutrino masses, so the breaking of dilatation symmetry could be the origin of the see-saw scale.
The renormalization group flow is driven toward $Λ_{ph} = 0$ in the far infrared when the beta function is positive, which the authors suggest could explain the approach to vanishing vacuum energy.

Reading between the lines

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

If the running between neutrino thresholds really sets the observed vacuum energy, then the cosmological constant is not a constant at all but a quantity that redshifted during cosmic history, which would leave a mild imprint on the expansion history at epochs between neutrino decoupling and today.
The required near-equality $m_S ≳ 4 m_ν$ between a new scalar and the lightest neutrino is unexplained in the paper; a natural reading is that both masses descend from the same intermediate scale $M$, which would make the submillimetre force range and the neutrino mass hierarchy two faces of one parameter.
Short-range gravity experiments that improve sensitivity at $0.02$ to $0.2$ mm with composition-differing test masses are the cheapest route to confirm or exclude the proposal, since the predicted coupling strength of about $1/3$ of gravity is well inside existing torsion-balance reach.
The same machinery applied at the Fermi scale would generate a far larger contribution once the top quark and weak bosons become active, so the paper's own logic implies that the cancellation must be re-established at each threshold, a requirement the authors defer to future work.

Original abstract

A large value of the cosmological constant (CC) is induced in the Standard Model (SM) of Elementary Particle Physics because of Spontaneous Symmetry Breaking. To provide a small value of the observable CC one has to introduce the vacuum term which cancels the induced one at some point in the very far infrared cosmic scale. Starting from this point we investigate whether the cancellation is preserved at different energy scales. We find that the running of the Higgs mass, couplings and the vacuum term inevitably result in a scaling dependence of the observable CC value. As a consequence one meets a nonzero CC at an energy scale comparable to the typical electron neutrino mass suggested by some experiments, and the order of magnitude of this constant is roughly the one derived from recent supernovae observations. However the sign of it is negative -- opposite to what is suggested by these observations. This discrepancy may be a hint of the existence of an extra very light scalar, perhaps a Cosmon-like dilaton, which should essentially decouple from the SM Lagrangian, but that it nevertheless could mediate new macroscopic forces in the submillimeter range.

The compiled record

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

A large value of the cosmological constant (CC) is induced in the Standard Model (SM) of Elementary Particle Physics because of Spontaneous Symmetry Breaking. To provide a small value of the observable CC one has to introduce the vacuum term which cancels the induced one at some point in the very far infrared cosmic scale. Starting from this point we investigate whether the cancellation is preserved at different energy scales. We find that the running of the Higgs mass, couplings and the vacuum term inevitably result in a scaling dependence of the observable CC value. As a consequence one meets a nonzero CC at an energy scale comparable to the typical electron neutrino mass suggested by some experiments, and the order of magnitude of this constant is roughly the one derived from recent supernovae observations. However the sign of it is negative -- opposite to what is suggested by these observations. This discrepancy may be a hint of the existence of an extra very light scalar, perhaps a Cosmon-like dilaton, which should essentially decouple from the SM Lagrangian, but that it nevertheless could mediate new macroscopic forces in the submillimeter range.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A large value of the cosmological constant (CC) is induced in the Standard Model (SM) of Elementary Particle Physics because of Spontaneous Symmetry Breaking.	background 1.00	0.08	0.03	asserted 1.00
s1	To provide a small value of the observable CC one has to introduce the vacuum term which cancels the induced one at some point in the very far infrared cosmic scale.	background 0.93	0.11	0.04	asserted 0.98
s2	Starting from this point we investigate whether the cancellation is preserved at different energy scales.	contribution 0.85	0.40	0.02	unclear 0.55
s3	We find that the running of the Higgs mass, couplings and the vacuum term inevitably result in a scaling dependence of the observable CC value.	result 1.00	0.94	0.28	asserted 1.00
s4	As a consequence one meets a nonzero CC at an energy scale comparable to the typical electron neutrino mass suggested by some experiments, and the order of magnitude of this constant is roughly the one derived from recent supernovae observations.	result 0.99	0.82	0.22	asserted 0.86
s5	However the sign of it is negative -- opposite to what is suggested by these observations.	result 0.95	0.83	0.14	asserted 1.00
s6	This discrepancy may be a hint of the existence of an extra very light scalar, perhaps a Cosmon-like dilaton, which should essentially decouple from the SM Lagrangian, but that it nevertheless could mediate new macroscopic forces in the submillimeter range.	interpretation 1.00	0.35	0.69	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.88.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
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references and citing sentences
relations to other papers

levels run

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

References

Works cited (24)

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S. Weinberg, Rev. Mod. Phys., 61 (1989) 1, and references therein

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How the literature uses it

32 arXiv papers cite this in 62 sentences

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

Papers citing it · 43 sentences without a confident label

Contests it 1 paper

The corresponding authors claim that in the work on the RGE running of the CC [ref] part of the one-loop corrections to the CC have been missed out, namely the explicit $μ$ dependent part of the full one-loop effective action.

Relaxing neutrino mass bounds by a running cosmological constant · Conclusions and Outlook · contest 0.82

Cites it as support 1 paper

This interaction between the KR axion and gravitational Chern-Simons terms proves essential in leading [ref] to a cosmological vacuum of the RVM type [ref] , which in turn implies an early inflationary era, without inflaton fields [ref] .

Torsion in string-inspired cosmologies and the universe dark sector · Introduction: A case for going beyond GR and Riemannian Geometry · 0.99

Cites it as background 16 papers

We mention at this point that one such theoretical framework is the so-called “running vacuum model (RVM)” framework [ref] (where a time dependent vacuum energy is assumed, but with equation of state $w_vacuum = -1$ as in the de Sitter case).

Torsion in string-inspired cosmologies and the universe dark sector · Introduction: A case for going beyond GR and Riemannian Geometry · 0.80
The RVM cosmology [ref] is an effective cosmological framework, with a cosmic-time-varying dark energy $Λ(t)$, which, nonetheless, is still characterised by an equation of state of de Sitter type: [equation] where $p$ ($ρ$) denotes the vacuum pressure (energy) density.

Lorentz Symmetry Violation in String-Inspired Effective Modified Gravity Theories · Condensates and Running-Vacuum-Model inflation · 0.93
The fast and fruitful theoretical developments undergone by the RVM framework over time can easily be traced in the literature by considering the derivation of its basic equations for the vacuum energy density (VED), originally guessed from old semiqualitative arguments based on the renormalization group (RG) approach to cosmology – see [ref] and references therein – until reaching its mature status at present through explicit renormalization calculations within QFT in curved spacetime [ref] .

Running Vacuum in the expanding Universe: a unified QFT paradigm for Inflation and Dark Energy · Discussion and outlook · 0.88

Discussion

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

[1] [1]

S. Weinberg, Rev. Mod. Phys., 61 (1989) 1, and references therein

View on Pith 1989

[2] [2]

See e.g. W. Hollik and R.D. Peccei, in: Proc. of the 4 th Int. Symposium on Radiative Corrections (RADCOR 98) World Scientiﬁc 1999, Ed. J. Sol` a

View on Pith 1999

[3] [3]

See e.g. I.L. Buchbinder, S.D. Odintsov, I.L. Shapiro, Eﬀective Action in Quantum Gravity, IOP Publishing (Bristol, 1992)

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[4] [4]

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