Predicting the SUSY breaking scale in SUGRA models with degenerate vacua

A. W. Thomas; C. D. Froggatt; H. B. Nielsen; R. Nevzorov

REVIEW 3 major objections 4 minor 28 references

Predicting the SUSY breaking scale in SUGRA models with degenerate vacua

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper predicts that the measured dark energy density fixes the supersymmetry breaking scale between 20 and 400 TeV.

desk verdict Two-loop RG calculation is the real new content, but the two-loop correction changes the answer by two orders of magnitude, so the 20-400 TeV window is not a controlled prediction. read the letter →

arxiv 1909.02124 v2 pith:ACVK26DV submitted 2019-09-04 hep-ph hep-th

C. D. Froggatt , R. Nevzorov , H. B. Nielsen , A. W. Thomas This is my paper · ORCID

classification hep-phhep-th

keywords supersymmetrybreakingscalecosmologicalconstantdarkenergydensitydegeneratevacuaMultiplePointPrincipleN=1supergravitytwo-looprenormalizationgrouphiggsinomatter

topics Dark Matter

open problems Dark Matter Dark Energy

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

The paper argues that the observed dark energy density can be read as a measurement of the supersymmetry breaking scale. In $N=1$ supergravity with two exactly degenerate vacua—the physical vacuum and a supersymmetric Minkowski vacuum—the only energy in the second vacuum comes from dynamical supersymmetry breaking triggered by strong QCD and top-quark interactions near a scale $\Lambda_c$, giving $\rho_\Lambda \sim \Lambda_c^4$. Assuming the high-energy gauge and top-Yukawa couplings are the same in both vacua to within a few percent, two-loop renormalization group running yields $\Lambda_c \simeq 0.001$–$0.002$ eV precisely when the physical SUSY breaking scale $M_S$ lies between 20 and 400 TeV. If correct, this turns the cosmological constant from a fine-tuning nuisance into a boundary condition that fixes the superpartner mass scale.

What carries the argument

The central object is the Landau-pole scale $\Lambda_c$ of the supersymmetric vacuum, defined as the scale where the two-loop running of $\alpha_3$ and the top-quark Yukawa coupling $Y_t$ becomes singular. It carries the argument through the identification $\rho_\Lambda \sim \Lambda_c^4$ (Eq. 4) and the one-loop relation $\Lambda_c = M_S \exp[2\pi/(b_3 \alpha_3^{(2)}(M_S))]$ (Eq. 5), which ties the dark energy density to the SUSY breaking scale. The two-loop contribution matters because it substantially reduces the growth of $\alpha_3$ and $Y_t$ in the infrared, lowering $\Lambda_c$ into the sub-eV range. The paper also uses the matching conditions (8)–(9), which allow $\alpha_3^{(2)}(M_X)$ and $Y_t^{(2)}(M_X)$ to differ from their physical-vacuum values by $\pm3\%$, to produce the quoted $M_S$ window.

What would settle it

Discover a superpartner, such as a gluino or squark, with mass well below 20 TeV: this would directly falsify the lower end of the predicted $M_S$ window. Alternatively, compute the coefficient $C$ in $\rho_\Lambda = C\Lambda_c^4$ from first principles; if $C$ differs from 1 by an order of magnitude, the $M_S$ range predicted from the cosmological constant moves outside 20–400 TeV.

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Extended reading notes

Core claim

Under the Multiple Point Principle, the physical vacuum and a supersymmetric Minkowski vacuum are exactly degenerate in energy. In the second vacuum supersymmetry is broken dynamically when the strong coupling and top-quark Yukawa coupling run to a Landau pole at $\Lambda_c$, producing a vacuum energy density $\rho_\Lambda \sim \Lambda_c^4$ that is transferred to the physical vacuum. Evolving the couplings with two-loop renormalization group equations, and matching them at $M_X \simeq 2\times 10^{16}$ GeV up to $\pm3\%$ differences, the paper obtains $\Lambda_c \simeq 0.001$–$0.002$ eV when the physical SUSY breaking scale $M_S$ lies between 20 and 400 TeV. The paper further argues this interval is consistent with the upper bound on $M_S$ implied by the higgsino dark matter scenario.

Load-bearing premise

The prediction rests on the unproven assumption that the dark energy density in the supersymmetric vacuum equals $\Lambda_c^4$ up to an order-one coefficient; if that coefficient is not close to 1, the inferred range of $M_S$ shifts by an order of magnitude.

Editorial extensions

If this is right

If the prediction holds, the measured cosmological constant implies $M_S$ is too large for most sparticles to be produced at the LHC.
For $M_S \gtrsim 100$ TeV the gravitino is heavy enough to decay before Big Bang Nucleosynthesis, so the gravitino problem is avoided.
For $M_S$ near 20 TeV, the lightest sparticles can be considerably lighter than $M_S$ and may be within reach of the HE-LHC or FCC.
The derived $M_S$ interval is compatible with the higgsino dark matter requirement that $M_S \lesssim$ a few hundred TeV, making the degenerate-vacua and dark-matter arguments mutually consistent.
Because $\Lambda_c$ grows with $\alpha_3^{(2)}(M_X)$ and falls with $M_S$, precise measurements of the strong coupling at high energies would sharpen the predicted sparticle spectrum.

Reading between the lines

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

The paper does not derive the coefficient in $\rho_\Lambda = C\Lambda_c^4$; if a future calculation found $C$ an order of magnitude away from 1, the inferred $M_S$ window would shift correspondingly.
The effect is fragile to new physics: the paper itself notes that adding one $5+\bar{5}$ multiplet pair removes the Landau pole, so a discovery of new matter at low energies would eliminate this particular prediction.
The same degeneracy principle could be applied to other gauge groups or hidden sectors; the strength of the correlation between $\Lambda_c$ and the low-energy spectrum is a generic feature that could be tested elsewhere.
A lattice or three-loop computation of the Landau-pole position would sharpen the $M_S$ prediction from an order-of-magnitude range to a precise mass spectrum.

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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

The genuinely new content is the two-loop RG evolution of MSSM couplings in the second vacuum, which shifts the predicted SUSY scale from the earlier one-loop 10^3-10^4 TeV down to 20-400 TeV. The paper is transparent: it writes standard two-loop beta functions, integrates them numerically, and tabulates the Landau pole scale Lambda_c for M_S with a +/-3% spread on high-scale couplings. That is a legitimate, reproducible calculation, even without shipped code. The authors also explicitly acknowledge the main caveats, which earns credit.

The soft spot is load-bearing. Table 1 shows one- and two-loop determinations of Lambda_c differ by about two orders of magnitude at each M_S. That means the two-loop result is not a controlled estimate of the pole position. The pole sits where alpha_3 ~ O(1), so its location is scheme- and order-dependent. The exponential sensitivity in Eq. (5) turns a modest three-loop or threshold correction into a large shift in Lambda_c and, through rho_Lambda ~ Lambda_c^4, an order-of-magnitude change in the inferred M_S window. On top of that, the coefficient in Eq. (4) is set to O(1) with no derivation. Either issue alone would be a caveat; together they mean the 20-400 TeV range is not a stable quantitative prediction.

This is not a rejection. The calculation is explicit and honest. The conclusion that M_S should be in the hundreds of TeV, rather than 10^3-10^4 TeV, is a meaningful update to this program. But the precise interval is better read as a broad consistency constraint: if the degenerate-vacua framework is right, M_S sits somewhere in the multi-TeV to few-hundred-TeV neighborhood. The higgsino dark matter discussion is a side check and does not rescue the numerical instability.

The citation pattern is heavily self-referential, which is appropriate given the program's continuity. The paper deserves a serious referee; it is a coherent, openly hedged extension of a speculative idea with a new quantitative result. I would not cite it as a precision prediction, but it is worth engaging with as a consistency argument. Bring it to a reading group if you want an instructive example of a paper that presents a fragile calculation transparently while slightly overselling its precision.

Referee Report

3 major / 4 minor

Summary. The manuscript applies the Multiple Point Principle (MPP) to N=1 supergravity models with two degenerate vacua: a physical vacuum with broken supersymmetry and a supersymmetric Minkowski vacuum. Assuming that the gauge and top-quark Yukawa couplings are almost identical in the two vacua at high energies, the authors identify the dark energy density with the fourth power of the scale Λ_c at which non-perturbative strong interactions in the second vacuum are expected to trigger dynamical supersymmetry breaking. Using two-loop SM and MSSM renormalization-group equations, they compute Λ_c as a function of the SUSY breaking scale M_S in the physical vacuum and find that the measured cosmological constant is reproduced for M_S in the range 20–400 TeV. Section 4 argues that this range is consistent with an upper bound on M_S derived in the higgsino dark matter scenario.

Significance. If the degenerate-vacuum postulate is accepted, the paper offers a concrete and falsifiable link between the tiny cosmological constant and the scale of supersymmetry breaking. The two-loop RG computation is transparent, and the inclusion of the earlier one-loop results in Table 1 is a useful feature. The claimed 20–400 TeV window is interesting because it is consistent with the higgsino dark matter scenario while predicting that most sparticles are beyond LHC reach. However, the quantitative prediction is only as robust as two uncontrolled ingredients: the identification ρ_Λ ~ Λ_c^4 with an uncomputed O(1) coefficient, and the location of a Landau pole in a regime where the coupling is of order unity and the two-loop correction substantially changes the result. As it stands, the computation should be regarded as an order-of-magnitude consistency estimate rather than a precise prediction.

major comments (3)

[3, Table 1] The two-loop determination of Λ_c is not under perturbative control. The paper itself notes that the two-loop contributions substantially cancel the one-loop growth when α_3 ~ 1, and Table 1 shows that the one-loop and two-loop results differ by about two orders of magnitude (e.g., for M_S = 100 TeV, the bracketed one-loop interval is 0.027–1 eV versus the two-loop interval 1.7×10^-4–6.4×10^-3 eV). Because the location of a Landau pole at O(1) coupling is scheme- and order-dependent, and because no three-loop, threshold, or scheme-dependence estimate is given, the value of Λ_c used in the central numerical claim is not robust. The authors should either provide a quantitative estimate of the truncation uncertainty or explicitly reframe the 20–400 TeV statement as a broad consistency constraint rather than a prediction.
[2, Eq. (4)] The relation ρ_Λ ~ Λ_c^4 is written with a proportionality symbol and used as an equality with coefficient O(1) throughout the numerical analysis. No derivation or estimate of this coefficient is provided. Since the matching to the measured cosmological constant fixes the target Λ_c ~ 10^-3 eV only through this relation, a coefficient of 10 or 0.1 would shift the target and, through the exponential sensitivity exemplified in Eq. (5), would move the inferred M_S window substantially. The paper should state the assumed coefficient explicitly and quantify the sensitivity of Table 1 to it.
[3, Eq. (9) and Table 1] The width of the final M_S range (20–400 TeV) is largely determined by the hand-chosen ±3% variations of α_3^(2)(M_X) and Y_t^(2)(M_X) in Eq. (9). No physical mechanism or theoretical prior is given for these variations. If the variations were smaller, the M_S range would shrink; if larger, it would expand over orders of magnitude. The table should therefore be presented as a scan over assumed high-scale variations, and the sensitivity of the conclusion to the width in Eq. (9) should be shown explicitly.

minor comments (4)

[3, before Eq. (7)] The text says 'Assuming that tan β ≫ 1' and then later in the same paragraph restricts to 'tan β sufficiently small, i.e. tan β ≪ 50−60'. These statements are not contradictory but are easy to misread; please make the intended range unambiguous, e.g., 'tan β in the range roughly 10–50'.
[3, Table 1] The columns are ordered as M_S = 10^4, 100, 20, 400 TeV, which is non-monotonic and makes the trend harder to follow. Please reorder the columns as 20, 100, 400, 10^4 TeV.
[2, Eq. (5)] The one-loop expression for Λ_c uses b_3 without explicitly defining it in Eq. (5); the surrounding text later changes b_3 when extra 5+5 multiplets are added. A sentence defining b_3 for the pure MSSM and for the extended case would improve clarity.
[3, Eq. (10)] The 'infrared fixed point' values α_3 ≃ 6π/7 and, in the extended case, α_3 ≃ 1.15, Y_t ≃ 1.01 lie in a regime where the expansion parameter is not small; this should be acknowledged when Eq. (10) and Eq. (11) are used to infer the absence or presence of a Landau pole.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular step: the Λ_c–M_S relation is derived from measured low-energy couplings and inverted against the observed cosmological constant.

full rationale

The paper's central numerical claim is the relation between the SUSY-breaking scale M_S in the physical vacuum and the Landau-pole scale Λ_c in the second vacuum. This relation is constructed by integrating two-loop SM and MSSM beta functions starting from the external measured values M_t=173.3 GeV and α_3(M_Z)=0.118, imposing the matching condition α_3^{(2)}(M_S)=α_3^{(1)}(M_S) from Eq. (3), and then evolving Eqs. (6)–(7) to M_X. The resulting Λ_c(M_S) is then compared with the measured dark-energy density through ρ_Λ∼Λ_c^4 (Eq. (4)). Solving Λ_c(M_S)≈10^{-3} eV for M_S is an inversion of this RG-derived function, not a tautology: the output M_S is not an input except as the threshold location, and the mapping is determined by measured low-energy couplings. The MPP degeneracy postulate and the non-perturbative estimate ρ_Λ∼Λ_c^4 are assumptions, cited partly to previous work by the same authors (Refs. [9]–[10]), but they are not derived by reducing the target prediction to itself; they are physical postulates whose validity is a model assumption. The numerical sensitivity of Λ_c to higher-order corrections (the one-loop and two-loop entries in Table 1 differ by orders of magnitude) is a serious perturbative-control concern, but it is a correctness issue, not circularity. I therefore find no step in which a predicted quantity is identical by construction to a fitted input or to a self-cited uniqueness result.

Assumptions & free parameters 3 free parameters · 6 assumptions · 2 invented entities

The central calculation rests on a chain of unverified physical postulates: exact vacuum degeneracy, energy transfer from the second phase to the physical phase, a top quark condensate that breaks SUSY, and an O(1) coefficient in rho_Lambda = Lambda_c^4. The RG equations themselves are standard and taken from the literature. No new particle or force is introduced, but the second vacuum and condensate are postulated degrees of freedom that carry the entire explanation.

free parameters (3)

SUSY breaking scale M_S = 20-400 TeV
Scanned in Table 1; the interval is selected so that Lambda_c falls near 10^-3 eV, the fourth root of the measured dark energy density.
Coefficient C in rho_Lambda = C * Lambda_c^4 = 1 assumed
Eq. (4) is a dimensional estimate; no potential calculation fixes C, and the inferred M_S range shifts by orders of magnitude with C.
High-scale coupling variation delta = +/-3% for alpha_3 and Y_t at M_X
Eq. (9) sets this spread by hand; it determines the width of the Lambda_c intervals in Table 1 and therefore the breadth of the M_S prediction.

assumptions (6)

domain assumption Exact degeneracy of the physical and supersymmetric Minkowski vacua (MPP).
Postulated in Section 1; it sets the leading vacuum energy to zero and transfers the second-phase energy to the physical vacuum.
domain assumption High-scale couplings in the two vacua are almost identical, with a +/-3% allowed spread.
Section 3, Eqs. (8) and (9); no dynamical mechanism enforces this near-equality.
domain assumption Non-perturbative SU(3) dynamics in the second vacuum forms a top quark condensate that breaks SUSY.
Section 2; this converts the Landau pole scale Lambda_c into a vacuum energy density.
ad hoc to paper The vacuum energy density is rho_Lambda ~ Lambda_c^4 with an O(1) coefficient.
Eq. (4); the coefficient is not computed, and the predicted M_S range depends on it.
ad hoc to paper Two-loop beta functions remain meaningful up to the Landau pole.
Section 3 and Fig. 1; near alpha_3 ~ 1, higher-order corrections may be important.
standard math Standard two-loop RG equations for the SM and MSSM from Refs. [15] and [17].
The equations in (6) and (7) are taken from the literature and are standard.

invented entities (2)

Degenerate supersymmetric Minkowski vacuum (second phase)
purpose: Provides a zero-energy reference vacuum whose tiny non-perturbative energy is transferred to the physical vacuum by the MPP degeneracy assumption.
No observational handle; this is the central postulate that makes the cosmological constant calculation possible.
Top quark condensate in the second phase
purpose: Breaks SUSY dynamically and sets the vacuum energy scale through rho_Lambda ~ Lambda_c^4.
The paper infers the condensate scale from the Landau pole but does not compute the condensate or its potential.

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Cite this review

Pith. "Pith review of Predicting the SUSY breaking scale in SUGRA models with degenerate vacua." pith.science (2026). https://pith.science/paper/ACVK26DV

@misc{pith2026190902124,
  author       = {Pith},
  title        = {Pith review of: Predicting the SUSY breaking scale in SUGRA models with degenerate vacua},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ACVK26DV}},
  note         = {Machine review of arXiv:1909.02124}
}

read the original abstract

In N=1 supergravity the scalar potential may have supersymmetric (SUSY) and non-supersymmetric Minkowski vacua (associated with supersymmetric and physical phases) with vanishing energy density. In the supersymmetric Minkowski (second) phase some breakdown of SUSY may be induced by non-perturbative effects in the observable sector that give rise to a tiny positive vacuum energy density. Postulating the exact degeneracy of the physical and second vacua as well as assuming that at high energies the couplings in both phases are almost identical, one can estimate the dark energy density in these vacua. It is mostly determined by the SUSY breaking scale M_S in the physical phase. Exploring the two-loop renormalization group (RG) flow of couplings in these vacua we find that the measured value of the cosmological constant can be reproduced if M_S varies from 20 TeV to 400 TeV. We also argue that this prediction for the SUSY breaking scale is consistent with the upper bound on M_S in the higgsino dark matter scenario.

Figures

Figures reproduced from arXiv: 1909.02124 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

[1] [1]

D. L. Bennett, H. B. Nielsen, Int. J. Mod. Phys. A 9, 5155 (1994)

work page 1994

[2] [2]

D. L. Bennett, C. D. Froggatt, H. B. Nielsen, in Proceedings of the 27th International Conference on High energy Physics, Glasgow, Scotland, 1994 , p. 557; Perspectives in Particle Physics ’94, World Scientiﬁc, 1995 , p. 255, ed. D. Klabu˘ car, I. Picek and D. Tadi´ c [arXiv:hep-ph/9504294]

work page Pith review arXiv 1994

[3] [3]

C. D. Froggatt, H. B. Nielsen, Phys. Lett. B 368 (1996) 96

work page 1996

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C. D. Froggatt, L. V. Laperashvili, R. B. Nevzorov, H. B. Nielsen, M. Sher, hep- ph/0412333. C. D. Froggatt, L. Laperashvili, R. Nevzorov, H. B. Nielsen, M. Sher, Phys. Rev. D 73 (2006) 095005 [hep-ph/0602054]; C. D. Froggatt, R. Nevzorov, H. B. Nielsen, D. Thompson, Phys. Lett. B 657 (2007) 95 [arXiv:0708.2903 [hep-ph]]. J. McDowall, D. J. Miller, Front....

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Cosmological constant in SUGRA models and the multiple point principle

C. Froggatt, L. Laperashvili, R. Nevzorov and H. B. Nielsen, Phys. Atom. Nucl. 67 (2004) 582 [hep-ph/0310127]

work page Pith review arXiv 2004

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C. D. Froggatt, H. B. Nielsen, R. Nevzorov and A. W. Thomas, Int. J. Mod. Phys. A 32 (2017) 1730013 [arXiv:1704.08453 [hep-ph]]. 13

work page Pith review arXiv 2017

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No--scale supergravity and the Multiple Point Principle

C. Froggatt, L. Laperashvili, R. Nevzorov and H. B. Nielsen, Bled Workshops Phys. 5 (2004) no.2, 17 [hep-ph/0411273]

work page Pith review arXiv 2004